Skip to main content

Full text of "The scientific papers of James Clerk Maxwell"

See other formats


BOSTON  UNIVERSITY 
LIBRARIES 


m 


Mugar  Memorial  Library 


gAz^k^  IS^€eiAy'^lia'^c>i^Ai^, 


Qna^a/i/etl^t, 


^io^laM^-^  .S^-<s 


THE  SCIENTIFIC  PAPERS  OF 

JAMES  CLERK  MAXWELL 


Edited  by  W.  D.  NIVEN,  M.A.,  F.R.S. 


Two  Volumes  Bound  As  One 


DOVER   PUBLICATIONS,   INC.,   NEW  YORK 


All  rights  reserved  under  Pan  American  and  In- 
ternational Copyright  Conventions. 


Published  in  Canada  by  General  Publishing  Com- 
pany, Ltd.,  30  Lesmill  Road,  Don  Mills,  Toronto, 
Ontario. 

Published  in  the  United  Kingdom  by  Constable 
and  Company,  Ltd.,  10  Orange  Street,  London 
W.  C.  2. 


This  Dover  edition,  first  published  in  1965,  is  an 
unabridged  and  unaltered  republication  of  the  work 
first  pubhshed  by  Cambridge  University  Press  in 
1890.  This  edition  is  published  by  special  arrange- 
ment with  Cambridge  University  Press. 

The  work  was  originally  pubhshed  in  two  separate 
volumes,  but  is  now  published  in  two  volumes 
bound  as  one. 


Library  of  Congress  Catalog  Card  Number:  A53 -9813 


Manufactured  in  the  United  States  of  America 

Dover  Publications,  Inc. 

180  Varick  Street 
New  York,  N.  Y.  10014 


THE  SCIENTIFIC  PAPERS  OF 

JAMES  CLERK  MAXWELL 


Edited  by  W.  D.  NIVEN,  M.A.,  F.R.S, 


Volume  One 


TO    HIS    GRACE 

THE    DUKE    OF    DEVONSHIRE    K.G. 

CHANCELLOR    OF    THE    UNIVERSITY    OF    CAMBRIDGE 

FOUNDER    OF    THE    CAVENDISH    LABORATORY 

THIS    MEMORIAL    EDITION 

OF 

THE    SCIENTIFIC    PAPERS 

OF 

THE   FIRST   CAVENDISH   PROFESSOR   OF   EXPERIMENTAL   PHYSICS 

IS 

BY    HIS    GRACE'S    PERMISSION 

RESPECTFULLY    AND    GRATEFULLY    DEDICATED 


SHORTLY  after  the  death  of  Professor  James  Clerk  Maxwell  a  Committee  was 
formed,  consisting  of  graduate  members  of  the  University  of  Cambridge  and 
of  other  friends  and  admirers,  for  the  purpose  of  securing  a  fitting  memorial  of 
him. 

The  Committee  had  in  view  two  objects :  to  obtain  a  likeness  of  Professor 
Clerk  Maxwell,  which  should  be  placed  in  some  public  building  of  the  Uni- 
versity;  and  to  collect  and  publish  his  scattered  scientific  writings,  copies  of 
which,  so  far  as  the  funds  at  the  disposal  of  the  Committee  would  allow, 
should  be  presented  to  learned  Societies  and  Libraries  at  home  and  abroad. 

It  was  decided  that  the  likeness  should  take  the  form  of  a  marble  bust. 
This  was  executed  by  Sir  J.  E.  Boehm,  R.A.,  and  is  now  placed  in  the 
apparatus   room   of  the   Cavendish   Laboratory. 

In  carrying  out  the  second  part  of  their  programme  the  Committee 
obtained  the  cordial  assistance  of  the  Syndics  of  the  University  Press,  who 
willingly  consented  to  publish  the  present  work.  At  the  request  of  the  Syndics, 
Mr  W.  D.  Niven,  M.A.,  Fellow  and  Assistant  Tutor  of  Trinity  College  and 
now  Director  of  Studies  at  the  Royal  Naval  College,  Greenwich,  undertook  the 
duties  of  Editor. 

The  Committee  and  the  Syndics  desire  to  take  this  opportunity  of 
acknowledging  their  obligation  to  Messrs  Adam  and  Charles  Black,  Publishers 
of  the  ninth  Edition  of  the  EiicyclopcEdia  Biitannica,  to  Messrs  Taylor  and 
Francis,  Publishers  of  the  London,  Edinburgh,  and  Dublin  Philosophical  Maga- 
zine and  Journal  of  Science,  to  Messrs  Macmillan  and  Co.,  Publishers  of 
Nature  and  of  the  Cambridge  and  Dublin  Mathematical  Joui-nal,  to  Messrs 
Metcalfe  and  Co.,  Publishers  of  the  Quarterly  Journal  of  Pure  and  Applied 
Mathematics,  and  to  the  Lords  of  the  Conmiittee  of  Council  on  Education, 
Proprietors  of  the  Handbooks  of  the  South  Kensington  Museum,  for  their 
courteous  consent  to  allow  the  articles  which  Clerk  Maxwell  had  contributed  to 
these  publications  to  be  included  in  the  present  work ;  to  Mr  Norman  Lockyer 
for  the  assistance  which  he  rendered  in  the  selection  of  the  articles  re-printed 
from  Nature;  and  their  further  obligation  to  Messrs  Macmillan  and  Co.  for 
permission  to  use  in  this  work  the  steel  engravings  of  Faraday,  Clerk  Maxwell, 
and  Helmholtz  from  the  Nature  Series  of  Portraits. 


Numerous  and  important  Papers,  contributed  by  Clerk  Maxwell  to  the 
Transactions  or  Proceedings  of  the  Royal  Societies  of  London  and  of  Edinburgh, 
of  the  Cambridge  Philosophical  Society,  of  the  Royal  Scottish  Society  of  Arts, 
and  of  the  London  Mathematical  Society;  Lectures  delivered  by  Clerk  Maxwell 
at  the  Royal  Institution  of  Great  Britain  pubHshed  in  its  Proceedings;  as  well 
as  Communications  and  Addresses  to  the  British  Association  published  in  its 
Reports,  are  also  included  in  the  present  work  with  the  sanction  of  the  above 
mentioned  learned  bodies. 

The  Essay  which  gained  the  Adams  Prize  for  the  year  1856  in  the 
University  of  Cambridge,  the  introductory  Lecture  on  the  Study  of  Experimental 
Physics  delivered  in  the  Cavendish  Laboratory,  and  the  Rede  Lecture  delivered 
before  the  University  in  1878,  complete  this  collection  of  Clerk  Maxwell's  scientific 
writings. 

The  diagrams  in  this  work  have  been  re-produced  by  a  photographic 
process  from  the  original  diagrams  in  Clerk  Maxwell's  Papers  by  the  Cambridge 
Scientific   Instrument  Company. 

It  only  remams  to  add  that  the  footnotes  inserted  by  the  Editor  are 
enclosed  between  square   brackets. 

Cambridge,  Augv^t,  1890. 


PEEFACE. 


CLERK  MAXWELL'S  biography  has  been  written  by  Professors  Lewis  Campbell  and 
Wm.  Garnett  with  so  much  skill  and  appreciation  of  their  subject  that  nothing  further 
remains  to  be  told.  It  would  therefore  be  presumption  on  the  part  of  the  editor  of  his 
papers  to  attempt  any  lengthened  narrative  of  a  biographical  character.  At  the  same  time 
a  memorial  edition  of  an  author's  collected  writings  would  hardly  be  complete  without 
some  account  however  slight  of  his  life  and  works.  Accordingly  the  principal  events  of 
Clerk  Maxwell's  career  will  be  recounted  in  the  following  brief  sketch,  and  the  reader 
who  wishes  to  obtain  further  and  more  detailed  information  or  to  study  his  character  in 
its  social  relations  may  consult  the  interesting  work  to  which  reference  has  been  made. 

James  Clerk  Maxwell  was  descended  from  the  Clerks  of  Penicuick  in  Midlothian, 
a  well-known  Scottish  family  whose  history  can  be  traced  back  to  the  IGth  century.  The 
first  baronet  served  in  the  parliament  of  Scotland.  His  eldest  son,  a  man  of  learning, 
was  a  Baron  of  the  Exchequer  in  Scotland.  In  later  times  John  Clerk  of  Eldin  a 
member  of  the  family  claimed  the  credit  of  having  invented  a  new  method  of  breaking 
the  enemy's  line  in  naval  warfare,  an  invention  said  to  have  been  adopted  by  Lord 
Rodney  in  the  battle  which  he  gained  over  the  French  in  1782.  Another  John  Clerk, 
son  of  the  naval  tactitian,  was  a  lawyer  of  much  acumen  and  became  a  Lord  of  the 
Court  of  Session.  He  was  distinguished  among  his  Edinburgh  contemporaries  by  his  ready 
and  sarcastic  wit. 

The  father  of  the  subject  of  this  memoir  was  John,  brother  to  Sir  George  Clerk  of 
Penicuick.  He  adopted  the  surname  of  Maxwell  on  succeeding  to  an  estate  in  Kirkcud- 
brightshire which  came  into  the  Clerk  family  through  marriage  with  a  Miss  Maxwell.  It 
cannot  be  said  that  he  was  possessed  of  the  energy  and  activity  of  mind  which  lead 
to  distinction.  He  was  in  truth  a  somewhat  easy-going  but  shrewd  and  intelligent 
man,  whose  most  notable  characteristics  were  his  perfect  sincerity  and  extreme  benevolence. 
He  took  an  enlightened  interest  in  mechanical  and  scientific  pursuits  and  was  of  an 
essentially  practical  turn  of  mind.  On  leaving  the  University  he  had  devoted  himself 
to  law  and  was  called  to  the  Scottish  Bar.  It  does  not  appear  however  that  he  met 
mth   any   great    success    in    that    profession.      At    all    events,   a    quiet    life    in    the    country 


X  PREFACE. 

presented  so  many  attractions  to  his  wife  as  well  as  to  himself  that  he  was  easily  induced 
to  relinquish  his  prospects  at  the  bar.  He  had  been  married  to  Frances,  daughter  of 
Robert  Cay  of  N.  Charlton,  Northumberland,  a  lady  of  strong  good  sense  and  resolute 
character. 

The  country  house  which  was  their  home  after  they  left  Edinburgh  was  designed 
by  John  Clerk  Maxwell  himself  and  was  built  on  his  estate.  The  house,  which  was  named 
Glenlair,  was  surrounded  by  fine  scenery,  of  which  the  water  of  Urr  with  its  rocky  and 
wooded  banks  formed  the  principal  charm. 

James  was  bom  at  Edinburgh  on  the  13th  of  June,  1831,  but  it  was  at  Glenlair 
that  the  greater  part  of  his  childhood  was  passed.  In  that  pleasant  spot  under  healthful 
influences  of  all  kinds  the  child  developed  into  a  hardy  and  ccirageous  boy.  Not 
precociously  clever  at  books  he  was  yet  not  without  some  signs  of  future  intellectual 
strength,  being  remarkable  for  a  spirit  of  inquiry  into  the  caupjs  and  connections  of  the 
phenomena  around  him.  It  was  remembered  afterwards  when  he  had  become  distinguished, 
that  the  questions  he  put  as  a  child  shewed  an  amount  of  thoughtfulness  which  for  his 
years  was  very  unusual. 

At  the  age  of  ten,  James,  who  had  lost  his  mother,  was  placed  under  the  charge  of 
relatives  in  Edinburgh  that  he  might  attend  the  Edinburgh  Academy.  A  charming  account 
of  his  school  days  is  given  in  the  narrative  of  Professor  Campbell  who  was  Maxwell's 
schoolfellow  and  in  after  life  an  intimate  friend  and  constant  correspondent.  The  child  is 
father  to  the  man,  and  those  who  were  privileged  to  know  the  man  Maxwell  will  easily 
recognise  Mr  Campbell's  picture  of  the  boy  on  his  first  appearance  at  school, — the  home- 
made garments  more  serviceable  than  fashionable,  the  rustic  speech  and  curiously  quaint 
but  often  humorous  manner  of  conveying  his  meaning,  his  bewilderment  on  first  undergoing 
the  routine  of  schoolwork,  and  his  Spartan  conduct  under  various  trials  at  the  hands  of 
his  schoolfellows.  They  will  further  feel  how  accurate  is  the  sketch  of  the  boy  become 
accustomed  to  his  surroundings  and  rapidly  assuming  the  place  at  school  to  which  his 
mental  powers  entitled  him,  while  his  superfluous  energy  finds  vent  privately  in  carrying 
out  mechanical  contrivances  and  geometrical  constructions,  in  reading  and  even  trying  his 
hand  at  composing  ballads,  and  in  sending  to  his  father  letters  richly  embellished  with 
grotesquely  elaborate  borders  and  drawings. 

An  event  of  his  school-days,  worth  recording,  was  his  invention  of  a  mechanical  method 
of  drawing  certain  classes  of  Ovals.  An  account  of  this  method  was  printed  in  the 
Proceedings  of  the  Royal  Society  of  Edinburgh  and  forms  the  first  of  his  writings 
collected  in  the  present  work.  The  subject  was  introduced  to  the  notice  of  the  Society 
by  the  celebrated  Professor  James  Forbes,  who  from  the  first  took  the  greatest  possible 
interest  in  Maxwell's  progress.  Professor  Tait,  another  schoolfellow,  mentions  that  at  the 
time  when  the  paper  on  the  Ovals  was  written.  Maxwell  had  received  no  instruction  in 
Mathematics  beyond  a  little  Euclid  and  Algebra. 


PREFACE.  aa 

In  1847  Maxwell  entered  the  University  of  Edinburgh  where  he  remained  for  three 
sessions.  He  attended  the  lectures  of  Kelland  in  Mathematics,  Forbes  in  Natural  Philosophy, 
Gregory  in  Chemistry,  Sir  W.  Hamilton  in  Mental  Philosophy,  Wilson  (Christopher  North) 
in  Moral  Philosophy.  The  lectures  of  Sir  W.  Hamilton  made  a  strong  impression  upon 
him,  in  stimulating  the  love  of  speculation  to  which  his  mind  was  prone,  but,  as  might 
have  been  expected,  it  was  the  Professor  of  Natural  Philosophy  who  obtained  the  chief  share 
of  his  devotion.  The  enthusiasm  which  so  distinguished  a  man  as  Forbes  naturally  inspired 
in  young  and  ardent  disciples,  evoked  a  feeling  of  personal  attachment,  and  the  Professor,  on 
his  part,  took  special  interest  in  his  pupil  and  gave  to  him  the  altogether  unusual 
privilege   of  working   with   his   fine   apparatus. 

What  was  the  nature  of  this  experimental  work  we  may  conjecture  from  a  perusal  of 
his  paper  on  Elastic  Solids,  written  at  that  time,  in  which  he  describes  some  experiments 
made  with  the  view  of  verifying  the  deductions  of  his  theory  in  its  application  to  Optics. 
Maxwell  would  seem  to  have  been  led  to  the  study  of  this  subject  by  the  following  cir- 
cumstance. He  was  taken  by  his  uncle  John  Cay  to  see  William  Nicol,  the  inventor  of 
the  polarising  prism  which  bears  his  name,  and  was  shewn  by  Nicol  the  colours  of  unan- 
nealed  glass  in  the  polariscope.  This  incited  Maxwell  to  study  the  laws  of  polarised  light 
and  to  construct  a  rough  polariscope  in  which  the  polariser  and  analyser  were  simple  glass 
reflectors.  By  means  of  this  instrument  he  was  able  to  obtain  the  colour  bands  of  unannealed 
glass.  These  he  copied  on  paper  in  water  colours  and  sent  to  Nicol.  It  is  gratifpng  to 
find  that  this  spirited  attempt  at  experimenting  on  the  part  of  a  mere  boy  was  duly 
appreciated  by  Nicol,  who  at  once  encouraged  and  delighted  him  by  a  present  of  a  couple  of 
his  prisms. 

The  paper  alluded  to,  viz.  that  entitled  "On  the  Equilibrium  of  Elastic  Solids,"  was 
read  to  the  Royal  Society  of  Edinburgh  in  1850.  It  forms  the  third  paper  which  Maxwell 
addressed  to  that  Society.  The  first  in  1846  on  Ovals  has  been  abready  mentioned.  The 
second,  under  the  title  "The  Theory  of  Rolling  Curves,"  was  presented  by  Kelland  in  1849. 

It  is  obvious  that  a  youth  of  nineteen  years  who  had  been  capable  of  these  efforts 
must  have  been  gifted  with  rare  originality  and  with  great  power  of  sustained  exertion. 
But  his  singular  self-concentration  led  him  into  habits  of  solitude  and  seclusion,  the  tendency 
of  which  was  to  confirm  his  peculiarities  of  speech  and  of  manner.  He  was  shy  and 
reserved  with  strangers,  and  his  utterances  were  often  obscure  both  in  substance  and  in 
his  manner  of  expressing  himself,  so  many  remote  and  unexpected  allusions  perpetually 
obtruding  themselves.  Though  really  most  sociable  and  even  fond  of  society  he  was 
essentially  reticent  and  reserved.  Mr  Campbell  thinks  it  is  to  be  regretted  that  Maxwell 
did  not  begin  his  Cambridge  career  eai'lier  for  the  sake  of  the  social  intercourse  which 
he  would  have  found  it  difficult  to  avoid  there.  It  is  a  question,  however,  whether  in 
losing  the  opportunity  of  using  Professor  Forbes'  apparatus  he  would  not  thereby  have  lost 
what   was  perhaps  the  most  valuable  part  of  his  early  scientific  training. 


XU  PREFACE. 

It  was  originally  intended  that  Maxwell  should  follow  his  father's  profession  of  advocate, 
but  this  intention  was  abandoned  as  soon  as  it  became  obvious  that  his  tastes  lay  in  a 
direction  so  decidedly  scientific.  It  was  at  length  determined  to  send  him  to  Cambridge 
and  accordingly  in  October,  1850,  he  commenced  residence  in  Peterhouse,  where  however  he 
resided  during  the  Michaelmas  Term  only.  On  December  14  of  the  same  year  he  migrated 
to  Trinity  College. 

It  may  readily  be  supposed  that  his  preparatory  training  for  the  Cambridge  course 
was  far  removed  from  the  ordinary  type.  There  had  indeed  for  some  time  been  practically 
no  restraint  upon  his  plan  of  study  and  his  mind  had  been  allowed  to  follow  its  natural 
bent  towards  science,  though  not  to  an  extent  so  absorbing  as  to  withdraw  him  from 
other  pursuits.  Though  he  was  not  a  sportsman, — indeed  sport  so  called  was  always  repugnant 
to  him — he  was  yet  exceedingly  fond  of  a  country  life.  He  was  a  good  horseman  and  a 
good  swimmer.  Whence  however  he  derived  his  chief  enjoyment  may  be  gathered  from  the 
account  which  Mr  Campbell  gives  of  the  zest  with  which  he  quoted  on  one  occasion  the 
lines  of  Bums  which  describe  the  poet  finding  inspiration  while  wandering  along  the  banks 
of  a  stream  in  the  free  indulgence  of  his  fancies.  Maxwell  was  not  only  a  lover  of  poetry 
but  himself  a  poet,  as  the  fine  pieces  gathered  together  by  Mr  Campbell  abundantly  testify. 
He  saw  however  that  his  true  calling  was  Science  and  never  regarded  these  poetical 
efforts  as  other  than  mere  pastime.  Devotion  to  science,  already  stimulated  by  successful 
endeavour,  a  tendency  to  ponder  over  philosophical  problems  and  an  attachment  to  English 
literature,  particularly  to  English  poetry, — these  tastes,  implanted  in  a  mind  of  singular 
strength  and  purity,  may  be  said  to  have  been  the  endowments  with  which  young  Maxwell 
began  his  Cambridge  career.  Besides  this,  his  scientific  reading,  as  we  may  gather  from  his 
papers  to  the  Royal  Society  of  Edinburgh  referred  to  above,  was  already  extensive  and 
varied.  He  brought  with  him,  says  Professor  Tait,  a  mass  of  knowledge  which  was  really 
immense  for  so  young  a  man  but  in  a  state  of  disorder  appalling  to  his  methodical 
private  tutor. 

Maxwell's  undergraduate  career  was  not  marked  by  any  specially  notable  feature.  His 
private  speculations  had  in  some  measure  to  be  laid  aside  in  favour  of  more  systematic 
study.  Yet  his  mind  was  steadily  ripening  for  the  work  of  his  later  years.  Among  those 
with  whom  he  was  brought  into  daily  contact  by  his  position,  as  a  Scholar  of  Trinity 
College,  were  some  of  the  brightest  and  most  cultivated  young  men  in  the  University.  In 
the  genial  fellowship  of  the  Scholars'  table  Maxwell's  kindly  humour  found  ready  play,  while 
in  the  more  select  coterie  of  the  Apostle  Club,  formed  for  mutual  cultivation,  he  found  a  field 
for  the  exercise  of  his  love  of  speculation  in  essays  on  subjects  beyond  the  lines  of  the 
ordinary  University  course.  The  composition  of  these  essays  doubtless  laid  the  foundation 
of  that  literary  finish  which  is  one  of  the  characteristics  of  Maxwell's  scientific  writings. 
His  biographers  have  preserved  several  extracts  on  a  variety  of  subjects  chiefly  of  a  specu- 
lative character.  They  are  remarkable  mainly  for  the  weight  of  thought  contained  in  them 
but  occasionally  also  for  smart  epigrams  and  for  a  vein  of  dry  and  sarcastic  humour. 


PREFACE. 


These  glimpses  into  Maxwell's  character  may  prepare  us  to  believe  that,  with  all  his 
shyness,  he  was  not  without  confidence  in  his  own  powers,  as  also  appears  from  the  account 
which  was  given  by  the  late  Master  of  Trinity  College,  Dr  Thompson,  who  was  Tutor  when 
Maxwell  personally  applied  to  him  for  permission  to  migrate  to  that  College.  He  appeared 
to  be  a  shy  and  diffident  youth,  but  presently  surprised  Dr  Thompson  by  producing  a 
bundle  of  papers,  doubtless  copies  of  those  we  have  already  mentioned,  remarking  "  Perhaps 
these  may  shew  you  that  I  am  not  unfit  to  enter  at  your  College." 

He  became  a  pupil  of  the  celebrated  William  Hopkins  of  Peterhouse,  under  whom  his 
course  of  study  became  more  systematic.  One  striking  characteristic  was  remarked  by  his 
contemporaries.  Whenever  the  subject  admitted  of  it  he  had  recourse  to  diagrams,  though 
his  fellow  students  might  solve  the  question  more  easily  by  a  train  of  analysis.  Many 
illustrations  of  this  manner  of  proceeding  might  be  taken  from  his  writings,  but  in 
truth  it  was  only  one  phase  of  his  mental  attitude  towards  scientific  questions,  which 
led  him  to  proceed  from  one  distinct  idea  to  another  instead  of  trusting  to  symbols  and 
equations. 

Maxwell's  published  contributions  to  Mathematical  Science  during  his  undergraduate  career 
were  few  and  of  no  great  importance.  He  found  time  however  to  carry  his  investigations 
into  regions  outside  the  prescribed  Cambridge  course.  At  the  lectures  of  Professor  Stokes* 
he  was  regular  in  his  attendance.  Indeed  it  appears  from  the  paper  on  Elastic  Solids, 
mentioned  above,  that  he  was  acquainted  with  some  of  the  writings  of  Stokes  before  he 
entered  Cambridge.  Before  1850,  Stokes  had  published  some  of  his  most  important  contri- 
butions to  Hydromechanics  and  Optics  ;  and  Sir  W.  Thomson,  who  was  nine  years'  Maxwell's 
senior  in  University  standing,  had,  among  other  remarkable  investigations,  called  special 
attention  to  the  mathematical  analogy  between  Heat-conduction  and  Statical  Electricity. 
There  is  no  doubt  that  these  authors  as  well  as  Faraday,  of  whose  experimental  researches 
he  had  made  a  careful  study,  exercised  a  powerful  directive  influence  on  his  mind. 

In  January,  1854,  Maxwell's  undergraduate  career  closed.  He  was  second  wrangler,  but 
shared  with  Dr  Routh,  who  was  senior  wrangler,  the  honours  of  the  First  Smith's  Prize. 
In  due  course  he  was  elected  Fellow  of  Trinity  and  placed  on  the  staff  of  College  Lecturers. 

No  sooner  was  he  released  from  the  restraints  imposed  by  the  Trinity  Fellowship 
Examination  than  he  plunged  headlong  into  original  work.  There  were  several  questions 
he  was  anxious  to  deal  with,  and  first  of  all  he  completed  an  investigation  on  the  Trans- 
formation of  Surfaces  by  Bending,  a  purely  geometrical  problem.  This  memoir  he  presentel 
to  the  Cambridge  Philosophical  Society  in  the  following  March.  At  this  period  he  also 
set  about  an  enquiry  into  the  quantitative  measurement  of  mixtures  of  colours  and  the 
causes  of  colour-blindness.  During  his  undergraduateship  he  had,  as  we  have  seen,  found 
time  for  the  study  of  Electricity.  This  had  already  borne  fruit  and  now  resulted  in  the 
first  of  his  important  memoirs  on  that  subject,— the  memoir  on  Faraday's  Lines  of  Force. 
•  Now  Sir  George  Gabriel  Stokes,  Bart.,  M.P.  for  the  University. 


Xiv  PREFACE. 

The  number  and  importance  of  his  papers,  published  in  1855—6,  bear  witness  to  his 
assiduity  during  this  period.  With  these  labours,  and  in  the  preparation  of  his  College 
lectures,  on  which  he  entered  with  much  enthusiasm,  his  mind  was  fully  occupied  and  the 
work  was  congenial.  He  had  formed  a  number  of  valued  friendships,  and  he  had  a  variety  of 
interests,  scientific  and  literary,  attaching  him  to  the  University.  Nevertheless,  when  the  chair 
of  Natural  Philosophy  in  Marischal  College,  Aberdeen,  fell  vacant,  Maxwell  became  a  candidate. 
This  step  was  probably  taken  in  deference  to  his  father's  wishes,  as  the  long  summer 
vacation  of  the  Scottish  College  would  enable  him  to  reside  with  his  father  at  Glenlair  for 
half  the  year  continuously.  He  obtained  the  professorship,  but  unhappily  the  kind  intentions 
which  prompted  him  to  apply  for  it  were  frustrated  by  the  death  of  his  father,  which  took 
place  in  April,  1856. 

It  is  doubtful  whether  the  change  from  the  Trinity  lectureship  to  the  Aberdeen 
professorship  was  altogether  prudent.  The  advantages  were  the  possession  of  a  laboratory  and 
the  long  uninterrupted  summer  vacation.  But  the  labour  of  drilling  classes  composed  chiefly 
of  comparatively  young  and  untrained  lads,  in  the  elements  of  mechanics  and  physics,  was 
not  the  work  for  which  Maxwell  was  specially  fitted.  On  the  other  hand,  in  a  large  college 
like  Trinity  there  could  not  fail  to  have  been  among  its  undergraduate  members,  some  of  the 
most  promising  young  mathematicians  of  the  University,  capable  of  appreciating  his  original 
genius  and  immense  knowledge,  by  instructing  whom  he  would  himself  have  derived  ad- 
vantage. 

In  1856  Maxwell  entered  upon  his  duties  as  Professor  of  Natural  Philosophy  at  Marischal 
College,  and  two  years  afterwards  he  married  Katharine  Mary  Dewar,  daughter  of  the 
Principal  of  the  College.  He  in  consequence  ceased  to  be  a  Fellow  of  Tiinity  College, 
but  was  afterwards  elected  an  honorary  Fellow,  at  the  same  time  as  Professor  Cayley. 

During  the  yeai*s  1856 — 60  he  was  still  actively  employed  upon  the  subject  of  colour 
sensation,  to  which  he  contributed  a  new  method  of  measurement  in  the  ingenious  instru- 
ment known  as  the  colour-box.  The  most  serious  demands  upon  his  powers  and  upon  his 
time  were  made  by  his  investigations  on  the  Stability  of  Saturn's  Rings.  This  was  the 
subject  chosen  by  the  Examiners  for  the  Adams  Prize  Essay  to  be  adjudged  in  1857,  and 
was  advertised  in  the  following  terms: — 

"The  Problem  may  be  treated  on  the  supposition  that  the  system  of  Rings  is 
exactly  or  very  approximately  concentric  with  Saturn  and  symmetrically  disposed  about 
the  plane  of  his  equator  and  different  hypotheses  may  be  made  respecting  the  physical 
constitution  of  the  Rings.  It  may  be  supposed  (1)  that  they  are  rigid;  (2)  that  they 
are  fluid  and  in  part  aeriform ;  (3)  that  they  consist  of  masses  of  matter  not  materially 
coherent.  The  question  will  be  considered  to  be  answered  by  ascertaining  on  these 
hypotheses  severally  whether  the  conditions  of  mechanical  stability  are  satisfied  by  the 
mutual  attractions  and  motions  of  the  Planet  and  the  Rings." 


PREFACE.  XV 

"It  is   desirable  that  an  attempt  should  also  be   made   to   determine  on  which  of 

the   above    hypotheses    the    appearances   both    of    the    bright    rings    and    the   recently 

discovered  dark   ring  may  be  most  satisfactorily  explained;   and  to  indicate  any  causes 

to  which  a  change  of  form  such  as  is  supposed  from  a  comparison  of  modem  with  the 

earlier  observations  to  have  taken  place,  may  be  attributed." 

It    is  sufficient   to   mention   here   that    Maxwell   bestowed  an  immense  amount  of   labour 

in   working    out    the    theory   as   proposed,   and    that   he   arrived   at   the   conclusion   that  "the 

only  system  of  rings  which  can  exist  is  one  composed  of  an  indefinite  number  of  unconnected 

particles    revolving    round    the    planet   with    different   velocities    according   to    their   respective 

distances.      These  particles   may  be  arranged  in  a  series  of   narrow  rings,  or  they  may  move 

about  through  each  other  irregularly.     In  the  first  case  the  destruction  of  the  system  will  be 

very  slow,  in    the    second   case  it  will   be  more  rapid,  but  there  may  be  a  tendency  towards 

an  aiTangement  in  narrow  rings  which  may  retard  the  process." 

Part  of  the  work,  dealing  with  the  oscillatory  waves  set  up  in  a  ring  of  satellites, 
was  illustrated  by  an  ingenious  mechanical  contrivance  which  was  greatly  admired  when 
exhibited  before  the  Royal  Society  of  Edinburgh. 

This  essay,  besides  securing  the  prize,  obtained  for  its  author  great  credit  among 
scientific  men.  It  was  characterized  by  Sir  George  Airy  as  one  of  the  most  remarkable 
applications  of  Mathematics  to  Physics  that  he  had  ever  seen. 

The  suggestion  has  been  made  that  it  was  the  irregular  motions  of  the  particles  which 
compose  the  Rings  of  Saturn  resulting  on  the  whole  in  apparent  regularity  and  uni- 
formity, which  led  Maxwell  to  the  investigation  of  the  Kinetic  Theory  of  Gases,  his  first 
contribution  to  which  was  read  to  the  British  Association  in  1859.  This  is  not  unlikely, 
but  it  must  also  be  borne  in  mind  that  Bernoulli's  Theory  had  recently  been  revived  by 
Herapath,  Joule  and  Clausius  whose  writings  may  have  drawn  Maxwell's  attention  to  the 
subject. 

In  1860  King's  College  and  Marischal  College  were  joined  together  as  one  institution, 
now  known  as  the  University  of  Aberdeen.  The  new  chair  of  Natural  Philosophy  thus 
created  was  filled  up  by  the  appointment  of  David  Thomson,  formerly  Professor  at  King's 
College  and  Maxwell's  senior.  Professor  Thomson,  though  not  comparable  to  Maxwell  as  a 
physicist,  was  nevertheless  a  remarkable  man.  He  was  distinguished  by  singular  force  of 
character  and  great  administrative  faculty  and  he  had  been  prominent  in  bringing  about 
the  fusion  of  the  Colleges.  He  was  also  an  admirable  lecturer  and  teacher  and  had  done 
much  to  raise  the  standard  of  scientific  education  in  the  north  of  Scotland.  Thus  the  choice 
made  by  the  Commissioners,  though  almost  inevitable,  had  the  effect  of  making  it  appear 
that  Maxwell  failed  as  a  teacher.  There  seems  however  to  be  no  evidence  to  support  such 
an  inference.  On  the  contrary,  if  we  may  judge  from  the  number  of  voluntary  students 
attending  his  classes  in  his  last  College  session,  he  would  seem  to  have  been  as  popular  as  a 
professor  as  he  was  personally  estimable. 


XVI  PREFACE. 

This  is  also  borne  out  by  the  fact  that  he  was  soon  afterwards  elected  Professor  of 
Natural  Philosophy  and  Astronomy  in  King's  College,  London.  The  new  appointment  had 
the  advantage  of  bringing  him  much  more  into  contact  with  men  in  his  own  department 
of  science,  especially  with  Faraday,  with  whose  electrical  work  his  own  was  so  intimately 
connected.  In  1862 — 63  he  took  a  prominent  part  in  the  experiments  organised  by  a 
Committee  of  the  British  Association  for  the  determination  of  electrical  resistance  in 
absolute  measure  and  for  placing  electrical  measurements  on  a  satisfactory  basis.  In  the 
experiments  which  were  conducted  in  the  laboratory  of  King's  College  upon  a  plan  due 
to  Sir  W.  Thomson,  two  long  series  of  measurements  were  taken  in  successive  years.  In 
the  first  year,  the  working  members  were  Maxwell,  Balfour  Stewart  and  Fleeming  Jenkin  ;  in 
the  second,  Charles  Hockin  took  the  place  of  Balfour  Stewart.  The  work  of  this  Committee 
was  communicated  in  the  form  of  reports  to  the  British  Association  and  was  afterwards 
republished  in  one  volume  by  Fleeming  Jenkin. 

Maxwell  was  a  professor  in  King's  College  from  1860  to  1865,  and  this  period  of  his 
life  is  distinguished  by  the  production  of  his  most  important  papers.  The  second  memoir 
on  Colours  made  its  appearance  in  1860.  In  the  same  year  his  first  papers  on  the  Kinetic 
Theory  of  Gases  were  published.  In  1861  came  his  papers  on  Physical  Lines  of  Force 
and  in  1864  his  greatest  memoii'  on  Electricity, — a  Dynamical  Theory  of  the  Electro- 
magnetic Field.  He  must  have  been  occupied  with  the  Dynamical  Theory  of  Gases  in  1865, 
as  two  important  papers  appeared  in  the  following  year,  first  the  Bakerian  lecture  on  the 
Viscosity  of  Gases,  and  next  the  memoir  on  the  Dynamical  Theory  of  Gases. 

The  mental  strain  involved  in  the  production  of  so  much  valuable  work,  combined 
with  the  duties  of  his  professorship  which  required  his  attention  during  nine  months  of 
the  year,  seems  to  have  influenced  him  in  a  resolution  which  in  1865  he  at  length 
adopted  of  resigning  his  chair  and  retiring  to  his  country  seat.  Shortly  after  this  he  had 
a  severe  illness.  On  his  recovery  he  continued  his  work  on  the  Dynamical  Theory  of 
Gases,  to  which  reference  has  just  been  made.  For  the  next  few  years  he  led  a  quiet 
and  secluded  life  at  Glenlair,  varied  by  annual  visits  to  London,  attendances  at  the  British 
Association  meetings  and  by  a  tour  in  Italy  in  1867.  He  was  also  Moderator  or  Examiner 
in  the  Mathematical  Tripos  at  Cambridge  on  several  occasions,  ofiBces  which  entailed  a  few 
weeks'  residence  at  the  University  in  winter.  His  chief  employment  during  those  years 
was  the  prepai-ation  of  his  now  celebrated  treatise  on  Electricity  and  Magnetism  which, 
however,  was  not  published  till  1873.  He  also  wrote  a  treatise  on  Heat  which  was 
published  in   1871. 

In  1871  Maxwell  was,  with  some  reluctance,  induced  to  quit  his  retreat  in  the 
country  and  to  enter  upon  a  new  career.  The  University  of  Cambridge  had  recently 
resolved  to  found  a  professorship  of  physical  science,  especially  for  the  cultivation  and 
teaching  of  the  subjects  of  Heat,  Electricity  and  Magnetism.  In  furtherance  of  this 
object  her  Chancellor,  the  Duke  of  Devonshire,  had  most  generously  undertaken  to  build 
a  laboratory  and   furnish   it  with   the   necessary   apparatus.     Maxwell   was   invited   to  fill  the 


PREFACE.  XVU 

new   chair   thus   formed    and    to   superintend    the    erection    of    the    laboratory.     In    October, 
1871,  he  delivered  his  inaugural  lecture. 

The  Cavendish  Laboratory,  so  called  after  its  founder,  the  present  venerable  chief  of 
the  family  which  produced  the  great  physicist  of  the  same  name,  was  not  completed 
for  practical  work  until  1874.  In  June  of  that  year  it  was  formally  presented  to  the 
University  by  the  Chancellor.  The  building  itself  and  the  fittings  of  the  several  rooms 
were  admirably  contrived  mainly  by  Maxwell  himself,  but  the  stock  of  apparatus  was 
smaller  than  accorded  with  the  generous  intentions  of  the  Chancellor.  This  defect  must 
be  attributed  to  the  anxiety  of  the  Professor  to  procure  only  instruments  by  the  best 
makers  and  with  such  improvements  as  he  could  himself  suggest.  Such  a  defect  therefore 
required  time  for  its  removal  and  afterwards  in  great  measure  disappeared,  apparatus  being 
constantly  added  to  the  stock  as  occasion  demanded. 

One  of  the  chief  tasks  which  Maxwell  undertook  was  that  of  superintending  and 
directing  the  energies  of  such  young  Bachelors  of  Arts  as  became  his  pupils  after 
having  acquired  good  positions  in  the  University  examinations.  Several  pupils,  who  have 
since  acquired  distinction,  carried  out  valuable  experiments  under  the  guidance  of  the 
Professor.  It  must  be  admitted,  however,  that  the  numbers  were  at  first  small,  but  perhaps 
this  was  only  to  be  expected  from  the  traditions  of  so  many  years.  The  Professor  was 
singularly  kind  and  helpful  to  these  pupils.  He  would  hold  long  conversations  with  them, 
opening  up  to  them  the  stores  of  his  mind,  giving  them  hints  as  to  what  they  might  try 
and  what  avoid,  and  was  always  ready  with  some  ingenious  remedy  for  the  experimental 
troubles  which  beset  them.  These  conversations,  always  delightful  and  instructive,  were, 
according  to  the  account  of  one  of  his  pupils,  a  liberal  education  in  themselves,  and  were 
repaid  in  the  minds  of  the  pupils  by  a  grateful  affection  rarely   accorded  to  any  teacher. 

Besides  discharging  the  duties  of  his  chair,  Maxwell  took  an  active  part  in  conducting 
the  general  business  of  the  University  and  more  particularly  in  regulating  the  courses  of 
study  in  Mathematics  and  Physics. 

For  some  years  previous  to  1866  when  Maxwell  returned  to  Cambridge  as  Moderator 
in  the  Mathematical  Tripos,  the  studies  in  the  University  had  lost  touch  with  the  great 
scientific  movements  going  on  outside  her  walls.  It  was  said  that  some  of  the  subjects  most 
in  vogue  had  but  little  interest  for  the  present  generation,  and  loud  complaints  began  to 
be  heard  that  while  such  branches  of  knowledge  as  Heat,  Electricity  and  Magnetism,  were 
left  out  of  the  Tripos  examination,  the  candidates  were  wasting  their  time  and  energy 
upon  mathematical  trifles  barren  of  scientific  interest  and  of  practical  results.  Into  the 
movement  for  reform  Maxwell  entered  warmly.  By  his  questions  in  1866  and  subsequent 
years  he  infused  new  life  into  the  examination ;  he  took  an  active  part  in  drafting  the 
new  scheme  introduced  in  1873 ;  but  most  of  all  by  his  writings  he  exerted  a  powerful 
influence  on  the  younger  members  of  the  University,  and  was  largely  instrumental  in 
bringing  about  the  change  which  has  been  now  effected. 


XVIU  PREFACE. 

In  the  first  few  years  at  Cambridge  Maxwell  was  busy  in  giving  the  final  touches 
to  his  great  work  on  Electricity  and  Magnetism  and  in  passing  it  through  the  press. 
This  work  was  published  in  1873,  and  it  seems  to  have  occupied  the  most  of  his  attention 
for  the  two  previous  years,  as  the  few  papers  published  by  him  during  that  period  relate 
chiefly  to  subjects  forming  part  of  the  contents.  After  this  publication  his  contributions  to 
scientific  journals  became  more  numerous,  those  on  the  Dynamical  Theory  of  Gases  being 
perhaps  the  most  important.  He  also  wrote  a  great  many  short  articles  and  reviews 
which  made  their  appearance  in  Nature  and  the  Encyclopcedia  Britannica.  Some  of  these 
essays  are  charming  expositions  of  scientific  subjects,  some  are  general  criticisms  of  the 
works  of  contemporary  writers  and  others  are  brief  and  appreciative  biographies  of  fellow 
workers  in  the  same  fields  of  research. 

An  undertaking  in  which  he  was  long  engaged  and  which,  though  it  proved  exceedingly 
interesting,  entailed  much  labour,  was  the  editing  of  the  "Electrical  Researches"  of  the  Hon. 
Henry  Cavendish.  This  work,  published  in  1879,  has  had  the  eflfect  of  increasing  the 
reputation  of  Cavendish,  disclosing  as  it  does  the  unsuspected  advances  which  that  acute 
physicist  had  made  in  the  Theory  of  Electricity,  especially  in  the  measurement  of  electrical 
quantities.  The  work  is  enriched  by  a  variety  of  valuable  notes  in  which  Cavendish's 
views  and  results  are  examined  by  the  light  of  modern  theory  and  methods.  Especially 
valuable  are  the  methods  applied  to  the  determination  of  the  electrical  capacities  of  con- 
ductors and  condensers,  a  subject  in  which  Cavendish  himself  shewed  considerable  skill 
both  of  a  mathematical  and  experimental  character. 

The  importance  of  the  task  undertaken  by  Maxwell  in  connection  with  Cavendish's 
papers  will  be  understood  from  the  following  extract  from  his  introduction  to  them. 

"It  is  somewhat  difficult  to  account  for  the  fact  that  though  Cavendish  had 
prepared  a  complete  description  of  his  experiments  on  the  charges  of  bodies,  and  had 
even  taken  the  trouble  to  write  out  a  fair  copy,  and  though  all  this  seems  to  have 
been  done  before  1774  and  he  continued  to  make  experiments  in  Electricity  till  1781 
and  lived  on  till  1810,  he  kept  his  manuscript  by  him  and  never  published  it." 

"Cavendish  cared  more  for  investigation  than  for  publication.  He  would  under- 
take the  most  laborious  researches  in  order  to  clear  up  a  difficulty  which  no  one 
but  himself  could  appreciate  or  was  even  aware  of,  and  we  cannot  doubt  that  the 
result  of  his  enquiries,  when  successful,  gave  him  a  certain  degree  of  satisfaction. 
But  it  did  not  excite  in  him  that  desire  to  communicate  the  discovery  to  others 
which  in  the  case  of  ordinary  men  of  science,  generally  ensures  the  publication  of 
their  results.  How  completely  these  researches  of  Cavendish  remained  unknown  to 
other  men  of  science  is  shewn  by  the  external  history  of  electricity." 

It  will  probably  be  thought  a  matter  of  some  difficulty  to  place  oneself  in  the 
position  of  a  physicist  of  a  century  ago  and  to  ascertain  the  exact  bearing  of  his 
experiments.     But   Maxwell   entered   upon   this  undertaking  with   the    utmost  enthusiasm  and 


PREFACE.  XIX 

succeeded  in  completely  identifying  himself  with  Cavendish's  methods.  He  shewed  that 
Cavendish  had  really  anticipated  several  of  the  discoveries  in  electrical  science  which  have  been 
made  since  his  time.  Cavendish  was  the  first  to  form  the  conception  of  and  to  measure 
Electrostatic  Capacity  and  Specific  Inductive  Capacity;  he  also  anticipated  Ohm's  law. 

The  Cavendish  papers  were  no  sooner  disposed  of  than  Maxwell  set  about  preparing 
a  new  edition  of  his  work  on  Electricity  and  Magnetism;  but  unhappily  in  the  summer 
term  of  1879  his  health  gave  way.  Hopes  were  however  entertained  that  when  he  returned 
to  the  bracing  air  of  his  country  home  he  would  soon  recover.  But  he  lingered  through 
the  summer  months  with  no  signs  of  improvement  and  his  spirits  gradually  sank  He  was 
finally  informed  by  his  old  fellow-student,  Professor  Sanders,  that  he  could  not  live  more 
than  a  few  weeks.  As  a  last  resort  he  was  brought  back  to  Cambridge  in  October  that  he 
might  be  under  the  charge  of  his  favourite  physician,  Dr  Paget*.  Nothing  however  could 
be  done  for  •  his  malady,  and,  after  a  painful  illness,  he  died  on  the  5th  of  November,  1879, 
in  his  49th   year. 

Maxwell  was  thus  cut  oflf  in  the  prime  of  his  powers,  and  at  a  time  when  the  depart- 
ments of  science,  which  he  had  contributed  so  much  to  develop,  were  being  every  day 
extended  by  fresh  discoveries.  His  death  was  deplored  as  an  irreparable  loss  to  science  and 
to  the  University,  in  which  his  amiable  disposition  was  as  universally  esteemed  as  his  genius 
was  admired. 

It  is  not  intended  in  this  preface  to  enter  at  length  into  a  discussion  of  the  relation 
which  Maxwell's  work  bears  historically  to  that  of  his  predecessors,  or  to  attempt  to  estimate 
the  effect  which  it  has  had  on  the  scientific  thought  of  the  present  day.  In  some  of  his 
papers  he  has  given  more  than  usually  copious  references  to  the  works  of  those  by  whom 
he  had  been  influenced;  and  in  his  later  papers,  especially  those  of  a  more  popular  nature 
which  appeared  in  the  Encyclopoedia  Britannica,  he  has  given  full  historical  outlines  of  some 
of  the  most  prominent  fields  in  which  he  laboured.  Nor  does  it  appear  to  the  present 
editor  that  the  time  has  yet  arrived  when  the  quickening  influence  of  Maxwell's  mind  on 
modem  scientific  thought  can  be  duly  estimated.  He  therefore  proposes  to  himself  the  duty 
of  recalling  briefly,  according  to  subjects,  the  most  important  speculations  in  which  Maxwell 
engaged. 

His  works  have  been  arranged  as  far  as  possible  in  chronological  order  but  they  fall 
naturally  under  a  few  leading  heads;  and  perhaps  we  shall  not  be  far  wrong  if  we  place 
first  in  importance  his  work  in  Electricity. 

His  first  paper  on  this  subject  bearing  the  title  "On  Faraday's  Lines  of  Force"  was 
read  before  the  Cambridge  Philosophical  Society  on  Dec.  11th,  1855.  He  had  been  previously 
attracted  by  Faraday's  method  of  expressing  electrical  laws,  and  he  here  set  before  himself 
the  task  of  shewing  that  the  ideas  which  had  guided  Faraday's  researches  were  not  incon- 
sistent with  the  mathematical  formulae  in  which  Poisson  and  others  had  cast  the  laws  of 
♦  Now  Sir  George  Edward  Paget,  K.C.B. 


PREFACE. 


Electricity.  His  object,  he  says,  is  to  find  a  physical  analogy  which  shall  help  the  mind 
to  grasp  the  results  of  previous  investigations  "without  being  committed  to  any  theory 
founded  on  the  physical  science  from  which  that  conception  is  borrowed,  so  that  it  is  neither 
draw  aside  from  the  subject  in  the  pursuit  of  analytical  subtleties  nor  carried  beyond  the 
truth  by  a  favorite  hypothesis." 

The  laws  of  electricity  are  therefore  compared  with  the  properties  of  an  incompressible 
fluid  the  motion  of  which  is  retarded  by  a  force  proportional  to  the  velocity,  and  the  fluid 
is  supposed  to  possess  no  inertia.  He  shews  the  analogy  which  the  lines  of  flow  of  such 
a  fluid  would  have  with  the  lines  of  force,  and  deduces  not  merely  the  laws  of  Statical 
Electricity  in  a  single  medium  but  also  a  method  of  representing  what  takes  place  when  the 
action  passes  from  one  dielectric  into  another. 

In  the  latter  part  of  the  paper  he  proceeds  to  consider  the  phenomena  of  Electro- 
magnetism  and  shews  how  the  laws  discovered  by  Ampere  lead  to  conclusions  identical  with 
those  of  Faraday.  In  this  paper  three  expressions  are  introduced  which  he  identifies  with 
the  components  of  Faraday's  electrotonic  state,  though  the  author  admits  that  he  has  not 
been  able  to  frame  a  physical  theory  which  would  give  a  clear  mental  picture  of  the 
various  connections  expressed  by  the  equations. 

Altogether  this  paper  is  most  important  for  the  light  which  it  throws  on  the  principles 
which  guided  Maxwell  at  the  outset  of  his  electrical  work.  The  idea  of  the  electrotonic 
state  had  afready  taken  a  firm  hold  of  his  mind  though  as  yet  he  had  formed  no  physical 
explanation  of  it.  In  the  paper  "On  Physical  Lines  of  Force"  printed  in  the  Philosophical 
Magazine,  Vol.  xxi.  he  resumes  his  speculations.  He  explains  that  in  his  former  paper  he 
had  found  the  geometrical  significance  of  the  Electrotonic  state  but  that  he  now  proposes 
"to  examine  magnetic  phenomena  from  a  mechanical  point  of  view."  Accordingly  he  propounds 
his  remarkable  speculation  as  to  the  magnetic  field  being  occupied  by  molecular  vortices, 
the  axes  of  which  coincide  with  the  lines  of  force.  The  cells  within  which  these  vortices 
rotate  are  supposed  to  be  separated  by  layers  of  particles  which  serve  the  double  purpose 
of  transmitting  motion  from  one  cell  to  another  and  by  their  own  motions  constituting  an 
electric  current.  This  theory,  the  parent  of  several  working  models  which  have  been  devised 
to  represent  the  motions  of  the  dielectric,  is  remarkable  for  the  detail  vnth  which  it  is 
worked  out  and  made  to  explain  the  various  laws  not  only  of  magnetic  and  electromagnetic 
action,  but  also  the  various  forms  of  electrostatic  action.  As  Maxwell  subsequently  gave  a 
more  general  theory  of  the  Electromagnetic  Field,  it  may  be  inferred  that  he  did  not  desire 
it  to  be  supposed  that  he  adhered  to  the  views  set  forth  in  this  paper  in  every  particular; 
but  there  is  no  doubt  that  in  some  of  its  main  features,  especially  the  existence  of 
rotation  round  the  lines  of  magnetic  force,  it  expressed  his  permanent  convictions.  In  his 
treatise  on  "Electricity  and  Magnetism,"  Vol.  ii.  p.  416,  (2nd  edition  427)  after  quoting  from 
Sir  W.  Thomson  on  the  explanation  of  the  magnetic  rotation  of  the  plane  of  the  polarisation 
of  light,  he  goes  on  to  say  of  the  present  paper, 


PREFACE.  XXI 

"A   theory   of  molecular    vortices   which    T    worked    out   at  considerable  length  was 
published  in  the  Phil.  Mag.  for  March,  April  and  May,  1861,  Jan.  and  Feb.  1862." 

- "  I  think  we  have  good  evidence  for  the  opinion  that  some  phenomenon  of  rotation 
is  going  on  in  the  magnetic  field,  that  this  rotation  is  performed  by  a  great  number 
of  very  small  portions  of  matter,  each  rotating  on  its  own  axis,  that  axis  being  parallel 
to  the  direction  of  the  magnetic  force,  and  that  the  rotations  of  these  various  vortices 
are  made  to  depend  on  one  another  by  means  of  some  mechanism  between  them." 

"The  attempt  which  I  then  made  to  imagine  a  working  model  of  this  mechanism 

must   be   taken   for   no   more   than   it   really   is,  a   demonstration   that   mechanism  may 

be   imagined   capable  of  producing  a  connection  mechanically  equivalent  to  the  actual 

connection  of  the  parts  of  the  Electromagnetic  Field." 

This   paper   is   also   important  as  containing  the  first  hint  of  the  Electromagnetic  Theory 

of    Light   which    was    to    be    more    fully    developed    afterwards    in    his    third    great    memoir 

"  On  the  Dynamical  Theory  of  the  Electromagnetic  Field."     This  memoir,  which  was  presented 

to   the   Royal   Society   on   the   27th  October,  1864,  contains  Maxwell's   mature   thoughts  on  a 

subject  which   had   so   long   occupied  his  mind.     It  was  afterwards  reproduced  in   his  Treatise 

with    trifling    modifications    in    the    treatment   of    its   parts,  but   without   substantial   changes 

in    its    main    features.      In    this    paper    Maxwell    reverses    the    mode    of    treating    electrical 

phenomena   adopted   by   previous   mathematical   writers;   for  while   they   had   sought   to   build 

up    the    laws    of    the    subject    by    starting    from    the    principles   discovered   by   Ampere,  and 

deducing    the    induction    of    currents    from   the   conservation    of    energy,   Maxwell   adopts   the 

method    of    first    arriving    at    the    laws    of    induction    and    then    deducing    the    mechanical 

attractions  and  repulsions. 

After  recalling  the  general  phenomena  of  the  mutual  action  of  cuiTents  and  magnets 
and  the  induction  produced  in  a  circuit  by  any  variation  of  the  strength  of  the  field  m 
which  it  lies,  the  propagation  of  light  through  a  luminiferous  medium,  the  properties  of 
dielectrics  and  other  phenomena  which  point  to  a  medium  capable  of  transmittmg  force 
and   motio^i,   he   proceeds. — 

"Thus    then   we   are   led   to   the   conception  of  a  complicated   mechanism   capable 

of  a   vast  variety  of  motions  but  at  the  same  time   so  connected   that   the   motion   of 

one   part  depends,  according  to  definite  relations,  on  the   motion  of  other   parts,   these 

teotions  being   communicated   by  forces   arising  from  the  relative  displacement  of  their 

connected  parts,    in    virtue    of   their    elasticity.      Such   a   mechanism   must   be   subject 

to  the  laws   of  Dynamics." 

On  applying  dynamical  principles  to  such  a  connected  system   he   attains  certain   general 

propositions    which,    on   being   compared    with   the   laws    of    induced   currents,    enable   him   to 

identify    certain   features   of  the    mechanism    with    properties    of  currents.      The   induction    of 

currehts  and  their  electromagnetic  attraction  are  thus  explained  and  connected. 


XXll  PREFACE. 

In  a  subsequent  part  of  the  memoir  he  proceeds  to  establish  from  these  premises 
the  general  equations  of  the  Field  and  obtains  the  usual  formulae  for  the  mechanical 
force  on  currents,  magnets  and  bodies  possessing  an  electrostatic  charge. 

He  also  returns  to  and  elaborates  more  fully  the  electromagnetic  Theory  of  Light. 
His  equations  shew  that  dielectrics  can  transmit  only  transverse  vibrations,  the  speed  of 
propagation  of  which  in  air  as  deduced  from  electrical  data  comes  out  practically  identical 
with  the  known  velocity  of  light.  For  other  dielectrics  the  index  of  refraction  is  equal 
to  the  square  root  of  the  product  of  the  specific  inductive  capacity  by  the  coefficient  of 
magnetic  induction,  which  last  factor  is  for  most  bodies  practically  unity.  Various  comparisons 
have  been  made  with  the  view  of  testing  this  deduction.  In  the  case  of  paraffin  wax  and 
some  of  the  hydrocarbons,  theory  and  experiment  agree,  but  this  is  not  the  case  with 
glass  and  some  other  substances.  Maxwell  has  also  applied  his  theory  to  media  which 
are  not  perfect  insulators,  and  finds  an  expression  for  the  loss  of  light  in  passing  through 
a  stratum  of  given  thickness.  He  remarks  in  confirmation  of  his  result  that  most  good 
conductors  are  opaque  while  insulators  are  transparent,  but  he  also  adds  that  electrolytes 
which  transmit  a  current  freely  are  often  transparent,  while  a  piece  of  gold  leaf  whose 
resistance  was  determined  by  Mr  Hockin  allowed  far  too  great  an  amount  of  light  to 
pass.  He  observes  however  that  it  is  possible  "there  is  less  loss  of  energy  when  the 
electromotive  forces  are  reversed  with  the  rapidity  of  light  than  when  they  act  for  sensible 
times  as  in  our  experiments."  A  similar  explanation  may  be  given  of  the  discordance 
between  the  calculated  and  observed  values  of  the  specific  inductive  capacity.  Prof.  J.  J, 
Thomson  in  the  Proceedings  of  the  Royal  Society,  Vol.  46,  has  described  an  experiment  by 
which  he  has  obtained  the  specific  inductive  capacities  of  various  dielectrics  when  acted 
on  by  alternating  electric  forces  whose  frequency  is  25,000,000  per  second.  He  finds  that 
under  these  conditions  the  specific  inductive  capacity  of  glass  is  very  nearly  the  same  as 
the  square  of  the  refractive  index,  and  very  much  less  than  the  value  for  slow  rates  of 
reversals.  In  illustration  of  these  remarks  may  be  quoted  the  observations  of  Prof.  Hertz  who 
has  shewn  that  vulcanite  and  pitch  are  transparent  for  waves,  whose  periods  of  vibration  are 
about  three  hundred  millionths  of  a  second.  The  investigations  of  Hertz  have  shewn  that 
electro-dynamic  radiations  are  transmitted  in  waves  with  a  velocity,  which,  if  not  equal  to,  is 
comparable  with  that  of  light,  and  have  thus  given  conclusive  proof  that  a  satisfactory 
theory  of  Electricity  must  take  into  account  in  some  form  or  other  the  action  of  the 
dielectric.  But  this  does  not  prove  that  Maxwell's  theory  is  to  be  accepted  in  every 
particular.  A  peculiarity  of  his  theory  is,  as  he  himself  points  out  in  his  treatise,  that 
the  variation  of  the  electric  displacement  is  to  be  treated  as  part  of  the  current  as  well 
as  the  current  of  conduction,  and  that  it  is  the  total  amount  due  to  the  sum  of  these 
which  flows  as  if  electricity  were  an  incompressible  fluid,  and  which  determines  external 
electrodynamic  actions.  In  this  respect  it  differs  from  the  theory  of  Helmholtz  which 
also  takes  into  account  the  action  of  the  dielectric.  Professor  J.  J.  Thomson  » in  his 
Review   of    Electric   Theories    has    entered    into    a    full    discussion    of   the    points    at    issue 


PREFACE.  XXlll 

between  the  two  above  mentioned  theories,  and  the  reader  is  referred  to  his  paper  for 
further  information  *.  Maxwell  in  the  memoir  before  us  has  also  applied  his  theory  to 
the  passage  of  light  through  crystals,  and  gets  rid  at  once  of  the  wave  of  normal  vibrations 
which  has  hitherto  proved  the  stumbling  block  in  other  theories  of  light. 

The  electromagnetic  Theory  of  Light  has  received  numerous  developments  at  the  hands 
of  Lord  Rayleigh,  Mr  Glazebrook,  Professor  J.  J.  Thomson  and  others.  These  volumes 
also  contain  various  shorter  papers  on  Electrical  Science,  though  perhaps  the  most  complete 
record  of  Maxwell's  work  in  this  department  is  to  be  found  in  his  Treatise  on  Electricity 
and  Magnetism  in  which  they  were  afterwards  embodied. 

Another  series  of  papers  of  hardly  less  importance  than  those  on  Electricity  are  the 
various  memoirs  on  the  Dynamical  Theory  of  Gases.  The  idea  that  the  properties  of 
matter  might  be  explained  by  the  motions  and  impacts  of  their  ultimate  atoms  is  as 
old  as  the  time  of  the  Greeks,  and  Maxwell  has  given  in  his  paper  on  "  Atoms "  a  full 
sketch  of  the  ancient  controversies  to  which  it  gave  rise.  The  mathematical  difficulties  of 
the  speculation  however  were  so  great  that  it  made  little  real  progress  till  it  was  taken 
up  by  Clausius  and  shortly  afterwards  by  Maxwell.  The  first  paper  by  Maxwell  on  the 
subject  is  entitled  "Illustrations  of  the  Dynamical  Theory  of  Gases"  and  was  published 
in  the  Philosophical  Magazine  for  January  and  July,  1860,  having  been  read  at  a  meeting 
of  the  British  Association  of  the  previous  year.  Although  the  methods  developed  in  this 
paper  were  afterwards  abandoned  for  others,  the  paper  itself  is  most  interesting,  as  it  indicates 
clearly  the  problems  in  the  theory  which  Maxwell  proposed  to  himself  for  solution,  and  so  far 
contains  the  germs  of  much  that  was  treated  of  in  his  next  memoir.  It  is  also  epoch-making, 
inasmuch  as  it  for  the  first  time  enumerates  various  propositions  which  ai-e  characteristic 
of  Maxwell's  work  in  this  subject.  It  contains  the  first  statement  of  the  distribution  of  velo- 
cities according  to  the  law  of  errors.  It  also  foreshadows  the  theorem  that  when  two  gases 
are  in  thermal  equilibrium  the  mean  kinetic  energy  of  the  molecules  of  each  system  is  the 
same ;    and   for   the  first   time   the  question  of  the  viscosity  of  gases  is  treated  dynamically. 

In  his  great  memoir  "On  the  Dynamical  Theory  of  Gases"  published  in  the  Philo- 
sophical Transactions  of  the  Royal  Society  and  read  before  the  Society  in  May,  1866,  he 
returns  to  this  subject  and  lays  down  for  the  first  time  the  general  d3niamical  methods 
appropriate  for  its  treatment.  Though  to  some  extent  the  same  ground  is  traversed  as  in 
his  former  paper,  the  methods  are  widely  different.  He  here  abandons  his  former  h}^othesis 
that  the  molecules  are  hard  elastic  spheres,  and  supposes  them  to  repel  each  other  with 
forces  varying  inversely  as  the  fifth  power  of  the  distance.  His  chief  reason  for  assuming 
this  law  of  action  appears  to  be  that  it  simplifies  considerably  the  calculation  of  the 
collisions  between  the  molecules,  and  it  leads  to  the  conclusion  that  the  coefficient  of 
viscosity  is  directly  proportional  to  the  absolute  temperature.  He  himself  undertook  an 
experimental  enquiry  for  the  purpose  of  verifying  this  conclusion,  and,  in  his  paper  on  the 
Viscosity  of  Gases,  he  satisfied  himself  of  its  correctness.     A  re-examination  of  the  numerical 

*  British  Association  Report,  1885. 


XXIV  PREFACE. 

reductions  made  in  the  course  of  his  work  discloses  however  an  inaccuracy  which  materially 
affects  the  values  of  the  coefl&cient  of  viscosity  obtained.  Subsequent  experiments  also  seem 
to  shew  that  the  concise  relation  he  endeavoured  to  establish  is  by  no  means  so  near 
the  truth  as  he  supposed,  and  it  is  more  than  doubtful  whether  the  action  between  two 
molecules  can  be  represented  by  any  law  of  so  simple  a  character. 

In  the  same  memoir  he  gives  a  fresh  demonstration  of  the  law  of  distribution  of 
velocities,  but  though  the  method  is  of  permanent  value,  it  labours  under  the  defect  of 
assuming  that  the  distribution  of  velocities  in  the  neighbourhood  of  a  point  is  the  same 
in  every  direction,  whatever  actions  may  be  taking  place  within  the  gas.  This  flaw  in 
the  argument,  first  pointed  out  by  Boltzmann,  seems  to  have  been  recognised  by  Maxwell, 
who  in  his  next  paper  "On  the  Stresses  in  Rarefied  Gases  arising  from  inequalities  of 
Temperature,"  published  in  the  Philosophical  Transactions  for  1879,  Part  I.,  adopts  a  form 
of  the  distribution  function  of  a  somewhat  different  shape.  The  object  of  this  paper  was 
to  arrive  at  a  theory  of  the  effects  observed  in  Crookes's  Radiometer.  The  results  of  the 
investigation  are  stated  by  Maxwell  in  the  introduction  to  the  paper,  from  which  it  would 
appear  that  the  observed  motion  cannot  be  explained  on  the  Dynamical  Theory,  unless  it 
be  supposed  that  the  gas  in  contact  with  a  solid  can  slide  along  the  surface  with  a  finite 
velocity  between  places  whose  temperatures  are  different.  In  an  appendix  to  the  paper 
he  shews  that  on  certain  assumptions  regarding  the  nature  of  the  contact  of  the  solid 
and  gas,  there  will  be,  when  the  pressure  is  constant,  a  flow  of  gas  along  the  surface 
from  the  colder  to  the  hotter  parts.  The  last  of  his  longer  papers  on  this  subject  is 
one  on  Boltzmann's  Theorem.  Throughout  these  volumes  will  be  found  numerous  shorter 
essays  on  kindred  subjects,  published  chiefly  in  Nature  and  in  the  Encyclopcedia  Britannica. 
Some  of  these  contain  more  or  less  popular  expositions  of  this  subject  which  Maxwell 
bad  himself  in  great  part  created,  while  others  deal  with  the  work  of  other  writers  in 
the  same  field.  They  are  profoundly  suggestive  in  almost  every  page,  and  abound  in  acute 
criticisms  of  speculations  which  he  could  not  accept.  They  are  always  interesting;  for 
although  the  larger  papers  are  sometimes  difficult  to  follow,  Maxwell's  more  popular  writings 
are  characterized  by  extreme  lucidity  and  simplicity  of  style. 

The  first  of  Maxwell's  papers  on  Colour  Perception  is  taken  from  the  Transactions  of 
the  Royal  Scottish  Society  of  Arts  and  is  in  the  form  of  a  letter  to  Dr  G.  Wilson  dated 
Jan.  4,  1855.  It  was  followed  directly  afterwards  by  a  communication  to  the  Royal  Society 
of  Edinburgh,  and  the  subject  occupied  his  attention  for  some  years.  The  most  important 
of  his  subsequent  work  is  to  be  found  in  the  papers  entitled  "An  account  of  Experiments 
on  the  Perception  of  Colour "  published  in  the  Philosophical  Magazine,  Vol  xiv.  and  "  On 
the  Theory  of  Compound  Colours  and  its  relation  to  the  colours  of  the  spectrum "  in  the 
Philosophical  Transactions  for  the  year  1860.  We  may  also  refer  to  two  lectures  delivered 
at  the  Royal  Institution,  in  which  he  recapitulates  and  enforces  his  main  positions  in  his 
usual  luminous  style.  Maxwell  from  the  first  adopts  Young's  Theory  of  Colour  Sensation, 
according  to   which   all   colours    may   ultimately   be    reduced    to    three,   a    red,   a   green    and 


PREFACE.  XXV 

a  violet.  This  theory  had  been  revived  by  Helmholtz  who  endeavoured  to  find  for  it  a 
physiological  basis.  Maxwell  however  devoted  himself  chiefly  to  the  invention  of  accurate 
methods  for  combining  and  recording  mixtures  of  colours.  His  first  method  of  obtaining 
mixtures,  that  of  the  Colour  Top,  is  an  adaptation  of  one  formerly  employed,  but  in 
Maxwell's  hands  it  became  an  instrument  capable  of  giving  precise  numerical  results  by 
means  which  he  added  of  varying  and  measuring  the  amounts  of  colour  which  were 
blended  in  the  eye.  In  the  representation  of  colours  diagrammatical ly  he  followed  Young 
in  employing  an  equilateral  triangle  at  the  angles  of  which  the  fundamental  colours  were 
placed.  All  colours,  white  included,  which  may  be  obtained  by  mixing  the  fundamental 
colours  in  any  proportions  will  then  be  represented  by  points  lying  within  the  triangle. 
Points  without  the  triangle  represent  colours  which  must  be  mixed  with  one  of  the  funda- 
mental tints  to  produce  a  mixture  of  the  other  two,  or  with  which  two  of  them  must  be 
mixed  to  produce  the  third. 

In  his  later  papers,  notably  in  that  printed  in  the  Philosophical  Transactions,  he 
adopts  the  method  of  the  Colour  Box,  by  which  different  parts  of  the  spectrum  may  be 
mixed  in  different  proportions  and  matched  with  white,  the  intensity  of  which  has  been 
suitably  diminished.  In  this  way  a  series  of  colour  equations  are  obtained  which  can  be 
used  to  evaluate  any  colour  in  terms  of  the  three  fundamental  colours.  These  observations 
on  which  Maxwell  expended  great  care  and  labour,  constitute  by  far  the  most  important 
data  regarding  the  combinations  of  colour  sensations  which  have  been  yet  obtained,  and 
are  of  permanent  value  whatever  theory  may  ultimately  be  adopted  of  the  physiology  of  the 
perception  of  colour. 

In  connection  with  these  researches  into  the  sensations  of  the  normal  eye,  may  be 
mentioned  the  subject  of  colour-blindness,  which  also  engaged  Maxwell's  attention,  and  is 
discussed  at  considerable  length  in  several  of  his  papers. 

Geometrical  Optics  was  another  subject  in  which  Maxwell  took  much  interest.  At  an  early 
period  of  his  career  he  commenced  a  treatise  on  Optics,  which  however  was  never  completed. 
His  first  paper  "On  the  general  laws  of  optical  instruments,"  appeared  in  1858,  but  a  brief 
account  of  the  first  part  of  it  had  been  previously  communicated  to  the  Cambridge  Philosophical 
Society.  He  therein  lays  down  the  conditions  which  a  perfect  optical  instrument  must  fulfil, 
and  shews  that  if  an  instrument  produce  perfect  images  of  an  object,  i.e.  images  free  from 
astigmatism,  curvature  and  distortion,  for  two  different  positions  of  the  object,  it  will  give 
perfect  images  at  all  distances.  On  this  result  as  a  basis,  he  finds  the  relations  between 
the  foci  of  the  incident  and  emergent  pencils,  the  magnifying  power  and  other  characteristic 
quantities.  The  subject  of  refraction  through  optical  combinations  was  afterwards  treated 
by  him  in  a  different  manner,  in  three  papers  communicated  to  the  London  Mathematical 
Society.  In  the  first  (1873),  "On  the  focal  lines  of  a  refracted  pencil,"  he  applies  Hamilton's 
characteristic  function  to  determine  the  focal  lines  of  a  thin  pencil  refracted  from  one 
isotropic   medium   into   another    at    any   surface    of    separation.      In    the   second    (1874),   "On 


XXVI  PREFACE. 

Hamilton's  characteristic  function  for  a  narrow  beam  of  light,"  he  considers  the  more  general 
question  of  the  passage  of  a  ray  from  one  isotropic  medium  into  another,  the  two  media 
being  separated  by  a  third  which  may  be  of  a  heterogeneous  character.  He  finds  the  most 
general  form  of  Hamilton's  characteristic  function  from  one  point  to  another,  the  first  being 
in  the  medium  in  which  the  pencil  is  incident  and  the  second  in  the  medium  in  which 
it  is  emergent,  and  both  points  near  the  principal  ray  of  the  pencil.  This  result  is  then 
applied  in  two  particular  cases,  viz.  to  determine  the  emergent  pencil  (1)  from  a  spectroscope, 
(2)  from  an  optical  instrument  symmetrical  about  its  axis.  In  the  third  paper  (1875)  he 
resumes  the  last-mentioned  application,  discussing  this  case  more  fully  under  a  somewhat 
simplified   analysis. 

It  may  be  remarked  that  all  these  papers  are  connected  by  the  same  idea,  which  was — 
first  to  study  the  optical  efiects  of  the  entire  instrument  without  examining  the  mechanism 
by  which  these  effects  are  produced,  and  then,  as  in  the  paper  in  1858,  to  supply  whatever 
data  may  be  necessary  by  experiments  upon  the  instrument  itself. 

Connected  to  some  extent  with  the  above  papers  is  an  investigation  which  was  published 
in  1868  "  On  the  cyclide."  As  the  name  imports,  this  paper  deals  chiefly  with  the  geometrical 
properties  of  the  surface  named,  but  other  matters  are  touched  on,  such  as  its  conjugate 
isothermal  functions.  Primarily  however  the  investigation  is  on  the  orthogonal  surfaces  to 
a  system  of  rays  passing  accurately  through  two  lines.  In  a  footnote  to  this  paper  Maxwell 
describes  the  stereoscope  which  he  invented  and  which  is  now  in  the  Cavendish   Laboratory. 

In  1868  was  also  published  a  short  but  important  article  entitled  "  On  the  best  arrange- 
ment for  producing  a  pure  spectrum  on  a  screen." 

The  various  papers  relating  to  the  stresses  experienced  by  a  system  of  pieces  joined 
together  so  as  to  form  a  frame  and  acted  on  by  forces  form  an  important  group  connected 
with  one  another.  The  first  in  order  was  "On  reciprocal  figures  and  diagrams  of  forces," 
published  in  1864.  It  was  immediately  followed  by  a  paper  on  a  kindred  subject,  "On 
the  calculation  of  the  equilibrium  and  stiffness  of  frames."  In  the  first  of  these  Maxwell 
demonstrates  certain  reciprocal  properties  in  the  geometry  of  two  polygons  which  are  related 
to  one  another  in  a  particular  way,  and  establishes  his  well-known  theorem  in  Graphical 
Statics  on  the  stresses  in  frames.  In  the  second  he  employs  the  principle  of  work  to 
problems  connected  with  the  stresses  in  frames  and  structures  and  with  the  deflections 
arising  from   extensions   in   any   of  the   connecting   pieces. 

A  third  paper  "  On  the  equilibrium  of  a  spherical  envelope,"  published  in  1867,  may 
here  be  referred  to.  The  author  therein  considers  the  stresses  set  up  in  the  envelope  by 
a  system  of  forces  applied  at  its  surface,  and  ultimately  solves  the  problem  for  two  normal 
forces  applied  at  any  two  points.  The  solution,  in  which  he  makes  use  of  the  principle 
of  inversion  as  it  is  applied  in  various  electrical  questions,  turns  ultimately  on  the  deter- 
mination  of  a   certain   function   first  introduced   by  Sir  George  Airy,  and   called   by  Maxwell 


PREFACE.  XXvii 

Airy's  Function  of  Stress.  The  methods  which  in  this  paper  were  attended  with  so  much 
success,  seem  to  have  suggested  to  Maxwell  a  reconsideration  of  his  former  work,  with  the 
view  of  extending  the  character  of  the  reciprocity  therein  established.  Accordingly  in  1870 
there  appeared  his  fourth  contribution  to  the  subject,  "On  reciprocal  figures,  frames  and 
diagrams  of  forces."  This  important  memoir  was  published  in  the  Transactions  of  the  Royal 
Society  of  Edinburgh,  and  its  author  received  for  it  the  Keith  Prize.  He  begins  with  a 
remarkably  beautiful  construction  for  drawing  plane  reciprocal  diagrams,  and  then  proceeds 
to  discuss  the  geometry  and  the  degrees  of  freedom  and  constraint  of  polyhedral  frames, 
his  object  being  to  lead  up  to  the  limiting  case  when  the  faces  of  the  polyhedron  become 
infinitely  small  and  form  parts  of  a  continuous  surface.  In  the  course  of  this  work  he 
obtains  certain  results  of  a  general  character  relating  to  inextensible  surfaces  and  certain 
otjiers  of  practical  utility  relating  to  loaded  frames.  He  then  attacks  the  general  problem  of 
representing  graphically  the  internal  stress  of  a  body  and  by  an  extension  of  the  meaning 
of  "Diagram  of  Stress,"  he  gives  a  construction  for  finding  a  diagram  which  has  mechanical 
as  well  as  geometrical  reciprocal  properties  with  the  figure  supposed  to  be  under  stress.  It 
is  impossible  with  brevity  to  give  an  account  of  this  reciprocity,  the  development  of  which 
in  Maxwell's  hands  forms  a  very  beautiful  example  of  analysis.  It  will  be  suflScient  to 
state  that  under  restricted  conditions  this  diagram  of  stress  leads  to  a  solution  for  the 
components  of  stress  in  terms  of  a  single  function  analogous  to  Airy's  Function  of  Stress. 
In  the  remaining  parts  of  the  memoir  there  is  a  discussion  of  the  equations  of  stress,  and 
it  is  shewn  that  the  general  solution  may  be  expressed  in  terms  of  three  functions  analogous 
to  Airy's  single  function  in  two  dimensions.  These  results  are  then  applied  to  special 
cases,  and  in  particular  the  stresses  in  a  horizontal  beam  with  a  uniform  load  on  its  upper 
surface   are   fully   investigated. 

On  the  subjects  in  which  Maxwell's  investigations  were  the  most  numerous  it  has 
been  thought  necessary,  in  the  observations  which  have  been  made,  to  sketch  out  briefly 
the  connections  of  the  various  papers  on  each  subject  with  one  another.  It  is  not  how- 
ever intended  to  enter  into  an  account  of  the  contents  of  his  other  contributions  to  science, 
and  this  is  the  less  necessary  as  the  reader  may  readily  obtain  the  information  he  may 
require  in  Maxwell's  own  language.  It  was  usually  his  habit  to  explain  by  way  of 
introduction  to  any  paper  his  exact  position  with  regard  to  the  subject  matter  and  to 
give  a  brief  account  of  the  nature  of  the  work  he  was  contributing.  There  are  however 
several  memoirs  which  though  unconnected  with  others  are  exceedingly  interesting  in  them- 
selves. Of  these  the  essay  on  Saturn's  Rings  will  probably  be  thought  the  most  important 
as  containing  the  solution  of  a  diflScult  cosmical  problem ;  there  are  also  various  papers  on 
Dynamics,  Hydromechanics  and  subjects  of  pure  mathematics,  which  are  most  useful  con- 
tributions  on   the   subjects   of  which   they   treat. 

The  remaining  miscellaneous  papers  may  be  classified  under  the  following  heads:  (a) 
Lectures  and  Addresses,  (b)  Essays  or  Short  Treatises,  (c)  Biographical  Sketches,  (d)  Criticisms 
and  Reviews. 


XXVIU  PREFACE. 

Class  (a)  comprises  his  addresses  to  the  British  Association,  to  the  London  Mathematical 
Society,  the  Rede  Lecture  at  Cambridge,  his  address  at  the  opening  of  the  Cavendish 
Laboratory  and  his  Lectures  at  the  Royal  Institution  and  to  the  Chemical  Society. 

Class  (6)  includes  all  but  one  of  the  articles  which  he  contributed  to  the  Encyclo- 
pcedia  Britanrdca  and  several  others  of  a  kindred  character  to  Nature. 

Class  (c)  contains  such  articles  as  "  Fai-aday "  in  the  Encyclopcedia  Britannica  and 
"  Helmholtz  "  in  Nature. 

Class  (d)  is  chiefly  occupied  with  the  reviews  of  scientific  books  as  they  were  pub- 
lished. These  appeared  in  Nature  and  the  most  important  have  been  reprinted  in  these 
pages. 

In  some  of  these  writings,  particularly  those  in  class  (b),  the  author  allowed  himself  a 
gi-eater  latitude  in  the  use  of  mathematical  symbols  and  processes  than  in  others,  as 
for  instance  in  the  article  "  Capillary  Attraction,"  which  is  in  fact  a  treatise  on  that  subject 
treated  mathematically.  The  lectures  were  upon  one  or  other  of  the  three  departments 
of  Physics  with  which  he  had  mainly  occupied  himself; — Colour  Perception,  Action  through 
a  Medium,  Molecular  Physics;  and  on  this  account  they  are  the  more  valuable.  In  the 
whole  series  of  these  more  popular  sketches  we  find  the  same  clear,  graceful  delineation  of 
principles,  the  same  beauty  in  arrangement  of  subject,  the  same  force  and  precision  in 
expounding  proofs  and  illustrations.  The  style  is  simple  and  singularly  free  fi-om  any  kind 
of  haze  or  obscurity,  rising  occasionally,  as  in  his  lectures,  to  a  strain  of  subdued  eloquence 
when   the   emotional    aspects   of  the   subject   overcome   the  purely  speculative. 

The  books  which  were  written  or  edited  by  Maxwell  and  published  in  his  lifetime  but 
which  are  not  included  in  this  collection  were  the  "Theory  of  Heat"  (1st  edition,  1871); 
"Electricity  and  Magnetism"  (1st  edition,  1873);  "The  Electrical  Researches  of  the  Hon- 
ourable Henry  Cavendish,  F.R.S.,  written  between  1771  and  1781,  edited  from  the  original 
manuscripts  in  the  possession  of  the  Duke  of  Devonshire,  K.G."  (1879).  To  these  may  be 
added  a  graceful  little  introductory  treatise  on  Dynamics  entitled  "Matter  and  Motion" 
(published  in  1876  by  the  Society  for  promoting  Christian  Knowledge).  Maxwell  also 
contributed  part  of  the  British  Association  Report  on  Electrical  Units  which  was  afterwards 
published  in  book  form  by  Fleeming  Jenkin. 

The  "Theory  of  Heat"  appeai-ed  in  the  Text  Books  of  Science  series  published  by 
Longmans,  Green  and  Co.,  and  was  at  once  hailed  as  a  beautiful  exposition  of  a  subject, 
part  of  which,  and  that  the  most  interesting  part,  the  mechanical  theory,  had  as  yet  but 
commenced  the  existence  which  it  owed  to  the  genius  and  laboui-s  of  Rankine,  Thomson  and 
Clausius.  There  is  a  certain  charm  in  Maxwell's  treatise,  due  to  the  freshness  and  originality 
of  its  expositions  which  has  rendered  it  a  great  favourite  with  students  of  Heat. 

After  his  death  an  "  Elementary  Treatise  on  Electricity,"  the  greater  part  of  which  he 
had   written,   was   completed    by   Professor   Garnett   and   published  in  1881.     The  aim  of  this 


PREFACE.  XXIX 

treatise  and   its  position   relatively   to   his   larger   work    may   be   gathered   from   the   following 
extract   from    Maxwell's  preface. 

"  In  this  smaller  book  I  have  endeavoured  to  present,  in  as  compact  a  form  as  I 
can,  those  phenomena  which  appear  to  throw  light  on  the  theory  of  electricity  and  to 
use  them,  each  in  its  place,  for  the  development  of  electrical  ideas  in  the  mind  of 
the   reader." 

"In  the  larger  treatise  I  sometimes  made  use  of  methods  which  I  do  not  think 
the  best  in  themselves,  but  without  which  the  student  cannot  follow  the  investigations 
of  the  founders  of  the  Mathematical  Theory  of  Electricity.  I  have  since  become  more 
convinced  of  the  superiority  of  methods  akic  to  those  of  Faraday,  and  have  therefore 
adopted   them    from   the   first." 

Of  the  "Electricity  and  Magnetism"  it  is  difficult  to  predict  the  future,  but  there  is 
no  doubt  that  since  its  publication  it  has  given  direction  and  colour  to  the  study  of 
Electrical  Science.  It  was  the  master's  last  word  upon  a  subject  to  which  he  had  devoted 
several  years  of  his  life,  and  most  of  what  he  wrote  found  its  proper  place  in  the  treatise. 
Several  of  the  chapters,  notably  those  on  Electromagnetism,  are  practically  reproductions  of 
his  memoirs  in  a  modified  or  improved  form.  The  treatise  is  also  remarkable  for  the  handling 
of  the  mathematical  details  no  less  than  for  the  exposition  of  physical  principles,  and  is 
enriched  incidentally  by  chapters  of  much  originality  on  mathematical  subjects  touched  on 
in  the  course  of  the  work.  Among  these  may  be  mentioned  the  dissertations  on  Spherical 
Harmonics  and  Lagrange's  Equations  in  Dj-namics. 

The  origin  and  growth  of  Maxwell's  ideas  and  conceptions  of  electrical  action,  cul- 
minating in  his  treatise  where  all  these  ideas  are  arranged  in  due  connection,  form  an 
interesting  chapter  not  only  in  the  history  of  an  individual  mind  but  in  the  history  of 
electrical  science.  The  importance  of  Faraday's  discoveries  and  speculations  can  hardly  be 
overrated  in  their  influence  on  Maxwell,  who  tells  us  that  before  he  began  the  study  of 
electricity  he  resolved  to  read  none  of  the  mathematics  of  the  subject  till  he  had  first 
mastered  the  "Experimental  Researches."  He  was  also  at  first  under  deep  obligations  to 
the  ideas  contained  in  the  exceedingly  important  papers  of  Sir  W.  Thomson  on  the  analogy 
between  Heat-Conduction  and  Statical  Electricity  and  on  the  Mathematical  Theory  of 
Electricity  in  Equilibrium.  In  his  subsequent  efforts  we  must  perceive  in  Maxwell,  possessed 
of  Faraday's  views  and  embued  with  his  spirit,  a  vigorous  intellect  bringing  to  bear  on  a 
subject  still  full  of  obscurity  the  steady  light  of  patient  thought  and  expending  upon  it 
all  the  resources  of  a  never  failing  ingenuity. 

Royal  Navax  College, 
Greenwich, 

August,  1890. 


TABLE    OF    CONTENTS. 


II. 
Ill 

IV. 

V. 

VI. 

VII. 

IX. 

X. 
XI. 

XII. 
XIII. 

XIV. 
XV. 


On  the  Description  of  Oval  Curves  and  those  having  a  plurality  of  Foci;   with 

remarks  by  Professor  Forbes 

On  the  Theory  of  Rolling  Curves ■* 

On  the  Equilibrium  of  Elastic  Solids ^^ 

Solutions  of  Problems 

On  the  Transformation  of  Surfaces  by  Bending 80 

On  a  paHicular  case  of  the  descent  of  a  heavy  body  in  a  resisting  medium        .       115 

On  the  Theory  of  Colours  in  relation  to  Colour- Blindness 119 

Experiments  on  Colour  as  perceived  by  the  Eye,  with  remarks  on  Colour -Blindness       126 

On  Faraday's  Lines  of  Force ^"^^ 

Description  of  a  New  Form  of  the  Platometer,  an  Instrument  for  measuring  the 

areas  of  Plane  Figures  drawn  on  paper 230 

On  the  elementary  theory  of  Optical  Instruments 238 

On  a  method   of  drawing  the   Theoretical  Form3  of  Faraday's  Lines   of  Force 

without  calculation 

On  the  unequal  sensibility  of  the  Foramen  Centrale  to  Light  of  different  Colours       242 
On   the   Theory  of  Compound  Colours  with  reference  to   mixtures  of  Blue   and 

Yellow  Light ^^'^ 

On  an  instrument  to  illustrate  Poimot's  TJieory  of  Rotation         .         .         •         .246 
On  a  Dynamical  Top,  for  exhibiting  the  phenomena  of  the  motions  of  a  body  of 
invariable  form  about  a  fixed  point,  with  s&ine  suggestions  as  to  the  Earth's 

motion 

Account  of  Experiments  on  the  Perception  of  Colour 263 

97 1 
On  the  general  laius  of  Optical  Instruments 

On  Theories  of  the  Constitution  of  Saturn's  Rings 286 

On  the  stability  of  the  motion  of  Saturn  s  Rings 288 

Illustrations  of  the  Dynamical  Theory  of  Gases 

On  the   Theory  of  Compound   Colours  and  the  Relations  of  the   Colours   of  the 
Spectrum 

On  the  Theory  of  Three  Primary  Colours ***^ 

451 

On  Physical  Lines  of  Force 

On  Reciprocal  Figures  and  Diagrams  of  Forces °^* 

A  Dynamical  Theory  of  the  Electromagnetic  Field 526 

On  the  Calculation  of  the  EquilibHum  and  Stiffness  of  Frames  ....       598 


ERRATA. 


Page  40.     In  the  first  of  equations  (12),  second  group  of  terms,  read 
(hP         dy'         d^ 


instead  of 


d^^^d^^^d^^ 


with  corresponding  changes  in  the  other  two  equations. 

Page  153,  five  lines  from  bottom  of  page,  read  127  instead  of  276 

Page  591,  four  lines  from  bottom  of  page  the  equation  should  be 

d^M     d2M_ldM 
da?  "^  db'      a  da~ 

Page  592,  in  the  first  line  of  the  expression  for  L  change 

-  K  cos  26    into     -  ^  cosec  26. 


[From  the  Proceedings  of  the  Royal  Society  of  Edinburgh,  Vol,  li.  April,  1846.] 


I.     On  the  Description  of  Oval  Curves,  and  those  having  a  plurality  of  Foci;  ivith 
remarks  by  Professor  Forbes.     Communicated  by  Professor  Forbes. 


Mr  Clerk  Maxwell  ingeniously  suggests  the  extension  of  the  common 
theory  of  the  foci  of  the  conic  sections  to  curves  of  a  higher  degree  of  com- 
plication in  the  following  manner : — 

(1)  As  in  the  ellipse  and  hyperbola,  any  point  in  the  curve  has  the 
sum  or  difference  of  two  lines  drawn  from  two  points  or  foci  =  a.  constant 
quantity,  so  the  author  infers,  that  curves  to  a  certain  degree  analogous,  may 
be  described  and  determined  by  the  condition  that  the  simple  distance  from 
one  focus  pliLS  a  multiple  distance  from  the  other,  may  be  =  a  constant  quantity; 
or  more  generally,   m  times  the  one  distance +  n  times  the  other  =  constant. 

(2)  The  author  devised  a  simple  mechanical  means,  by  the  wrapping 
of  a   thread   round  pins,  for   producing   these   curves.      See   Figs.   1    and   2.     He 


Fig.  1.     Two  FocL     Katios  1, 


Fig.  2.     Two  Foci     Ratios  2,  3. 


then    thought    of    extending    the    principle    to     other    curves,    whose    property 
should   be,   that   the  sum   of  the   simple   or    multiple   distances   of  any   point   of 


DESCRIPTION    OF    OVAL    CURVES. 


the  curve  from  three  or  more  points  or  foci,  should  be  =  a  constant  quantity ; 
and  this,  too,  he  has  effected  mechanically,  by  a  very  simple  arrangement  of 
a  string  of  given  length  passing  round  three  or  more  fixed  pins,  and  con- 
straining a  tracing   point,  P.     See  Fig.    3.     Farther,   the  author  regards   curves 

Fig.  3.     Three  Foci.     Eatios  of  Equality. 


of  the  first  kind  as  constituting  a  particular  class  of  curves  of  the  second 
kind,  two  or  more  foci  coinciding  in  one,  a  focus  in  which  two  strings  meet 
being  considered  a  double  focus;  when  three  strings  meet  a  treble  focus,  &c. 

Professor  Forbes  observed  that  the  equation  to  curves  of  the   first   class   is 
easily  found,  having  the  form 

V^+7=  a-VhJ{x-  c)'  +  y\ 

which  is  that  of  the  curve  known  under  the  name  of  the  First  Oval  of 
Descartes*.  Mr  Maxwell  had  already  observed  that  when  one  of  the  foci  was 
at  an  infinite  distance  (or  the  thread  moved  parallel  to  itself,  and  was  confined 
in  respect  of  length  by  the  edge  of  a  board),  a  curve  resembling  an  ellipse 
was  traced ;  from  which  property  Professor  Forbes  was  led  first  to  infer  the 
identity  of  the  oval  with  the  Cartesian  oval,  which  is  well  known  to  have  this 
property.  But  the  simplest  analogy  of  all  is  that  derived  from  the  method  of 
description,  r  and  r    being  the  radients  to  any  point  of  the  curve  from  the  two 

foci ; 

mr  +  nr  —  constant, 

which  in  fact  at  once  expresses  on  the  undulatory  theory  of  light  the  optical 
character  of  the  surface  in  question,  namely,  that  light  diverging  from  one 
focus   F  without  the   medium,   shall  be   correctly   convergent  at  another  point  / 

*  Herschel,  On  Light,  Art.  232 ;  Lloyd,  On  Light  and  Vision,  Chap.  vii. 


DESCRIPTION    OF    OVAL    CURVES.  J 

within   it ;    and   in    this   case   the   ratio   —   expresses   the   index   of  refraction   of 

the  medium*. 

If  we  denote  by  the  power  of  either  focus  the  number  of  strings  leading 
to  it  by  Mr  Maxwell's  construction,  and  if  one  of  the  foci  be  removed  to  an 
infinite  distance,  if  the  powers  of  the  two  foci  be  equal  the  curve  is  a  parabola ; 
if  the  power  of  the  nearer  focus  be  greater  than  the  other,  the  curve  is  an 
eUipse;  if  the  power  of  the  infinitely  distant  focus  be  the  greater,  the  curve 
is  a  hyperbola.  The  first  case  evidently  corresponds  to  the  case  of  the  reflection 
of  parallel  rays  to  a  focus,  the  velocity  being  unchanged  after  reflection;  the 
second,  to  the  refraction  of  parallel  rays  to  a  focus  in  a  dense  medium  (in 
which  light  moves  slower) ;   the  third  case  to  refraction  into  a  rarer  medium. 

The  ovals  of  Descartes  were  described  in  his  Geometry,  where  he  has  also 
given  a  mechanical  method  of  describing  one  of  themt,  but  only  in  a  particular 
case,  and  the  method  is  less  simple  than  Mr  Maxwell's.  The  demonstration  of 
the  optical  properties  was  given  by  Newton  in  the  Principia,  Book  i.,  prop.  97, 
by  the  law  of  the  sines;  and  by  Huyghens  in  1690,  on  the  Theory  of  Undu- 
lations in  his  Traite  de  la  Lumiere.  It  probably  has  not  been  suspected  that 
so  easy  and  elegant  a  method  exists  of  describing  these  curves  by  the  use  of 
a  thread  and  pins  whenever  the  powers  of  the  foci  are  commensurable.  For 
instance,  the  curve.  Fig.  2,  drawn  with  powers  3  and  2  respectively,  give  the 
proper  form  for  a  refracting  surface  of  a  glass,  whose  index  of  refraction  is  1'50, 
in  order  that  rays  diverging  from  f  may  be  refracted  to  F. 

As  to  the  higher  classes  of  curves  with  three  or  more  focal  points,  we 
cannot  at  present  invest  them  with  equally  clear  and  curious  physical  properties, 
but  the  method  of  drawing  a  curve  by  so  simple  a  contrivance,  which  shall 
satisfy  the  condition 

mr  +  nr  +pr"  +  &c.  =  constant, 

is  in  itself  not  a  little  interesting;  and  if  we  regard,  with  Mr  Maxwell,  the 
ovals  above  described,  as  the  limiting  case  of  the  others  by  the  coalescence 
of  two  or  more  foci,  we  have  a  farther  generalization  of  the  same  kind  as  that 
so  highly  recommended  by  Montucla^  by  which  Descartes  elucidated  the  conic 
sections  as  particular  cases  of  his  oval  curves. 

♦  This  was  perfectly  well  shewn  by  Hnyghens  in  his  Traite  de  la  Lumiere,  p.  111.     (1690.) 

+  Edit.  1683.     Geometria,  Lib.  ii.  p.  54. 

X  Histoire  dea  Mathematiqties.     First  Edit  IL  102. 


[From  the  Transactions  of  the  Royal  Society  of  Edinburgh,  Vol.  xvi.  Part  v.] 


II.     On   the    Theory   of  Rolling    Curves.      Communicated   by   the   Eev.    Professor 

Kelland. 


There  is  an  important  geometrical  problem  which  proposes  to  find  a  curve 
having  a  given  relation  to  a  series  of  curves  described  according  to  a  given 
law.     This  is  the  problem  of  Trajectories  in  its  general  form. 

The  series  of  curves  is  obtained  from  the  general  equation  to  a  curve  by 
the  variation  of  its  parameters.  In  the  general  case,  this  variation  may  change 
the  form  of  the  curve,  but,  in  the  case  which  we  are  about  to  consider,  the 
curve  is  changed  only  in  position. 

This  change  of  position  takes  place  partly  by  rotation,  and  partly  by  trans- 
ference through  space.  The  roUing  of  one  curve  on  another  is  an  example  of 
this  compound  motion. 

As  examples  of  the  way  in  which  the  new  curve  may  be  related  to  the 
series  of  curves,  we  may  take  the  following : — 

1.  The  new  curve  may  cut  the  series  of  curves  at  a  given  angle.  When 
this  angle  becomes  zero,  the  curve  is  the  envelope  of  the  series  of  curves. 

2.  It  may  pass  through  correspondiug  points  in  the  series  of  curves. 
There  are  many  other  relations  which  may  be  imagined,  but  we  shall  confine 
our  attention  to  this,  partly  because  it  aSbrds  the  means  of  tracing  various 
curves,  and  partly  on  account  of  the  connection  which  it  has  with  many 
geometrical  problems. 

Therefore  the  subject  of  this  paper  will  be  the  consideration  of  the  relations 
of  three  curves,  one  of  which  is  fixed,  while  the  second  rolls  upon  it  and 
traces  the  third.  The  subject  of  rolling  curves  is  by  no  means  a  new  one. 
The  first  idea  of  the  cycloid  is  attributed  to  Aristotle,  and  involutes  and 
evolutes  have  been  long  known. 


THE    THEORY    OF    ROLLING    CURVES.  0 

In  the  Histmy  of  the  Royal  Academy  of  Sciences  for  1704,  page  97, 
there  is  a  memoir  entitled  "Nouvelle  formation  des  Spirales,"  by  M.  Varignon, 
in  which  he  shews  how  to  construct  a  polar  curve  from  a  curve  referred  to 
rectangular  co-ordinates  by  substituting  the  radius  vector  for  the  abscissa,  and 
a  circular  arc  for  the  ordinate.  After  each  curve,  he  gives  the  curve  into 
which  it  is  "  unrolled,"  by  which  he  means  the  curve  which  the  spiral  must 
be  rolled  upon  in  order  that  its  pole  may  trace  a  straight  line;  but  as  this 
18  not  the  principal  subject  of  his  paper,  he  does  not  discuss  it  very  fully. 

There  is  also  a  memoir  by  M.  de  la  Hire,  in  the  volume  for  1706,  Part  ii., 
page  489,  entitled  "Methode  generale  pour  r^duire  toutes  les  Lignes  courbes  ^ 
des  Roulettes,  leur  generatrice  ou  leur  base  ^tant  donnde  telle  qu'on  voudra." 

M.  de  la  Hire  treats  curves  as  if  they  were  polygons,  and  gives  geome- 
trical constructions  for  finding  the  fixed  curve  or  the  rolling  curve,  the  other 
two  being  given;    but  he  does  not  work  any  examples. 

In  the  volume  for  1707,  page  79,  there  is  a  paper  entitled,  "Methode 
generale  pour  determiner  la  nature  des  Courbes  form^es  par  le  roulement  de 
toutes  sortes  de  Courbes  sur  une  autre  Courbe  quelconque."     Par  M.  Nicole. 

M.  Nicole  takes  the  equations  of  the  three  curves  referred  to  rectangular 
co-ordinates,  and  finds  three  general  equations  to  connect  them.  He  takes  the 
tracing-point  either  at  the  origin  of  the  co-ordinates  of  the  rolled  curve  or  not. 
He  then  shews  how  these  equations  may  be  simplified  in  several  particular 
cases.     These  cases  are — 

(1)  When  the  tracing-point  is  the  origin  of  the  roUed  curve. 

(2)  When  the  fixed  curve  is  the  same  as  the  rolling  cxirve. 

(3)  When  both  of  these  conditions  are  satisfied. 

(4)  When  the  fixed  line  is  straight. 

He  then  says,  that  if  we  roll  a  geometric  curve  on  itself,  we  obtain  a  new 
geometric  curve,  and  that  we  may  thus  obtain  an  infinite  number  of  geometric 
curves. 

The  examples  which  he  gives  of  the  application  of  his  method  are  all  taken 
from  the  cycloid  and  epicycloid,  except  one  which  relates  to  a  parabola,  rolling 
on  itself,  and  tracing  a  cissoid  with  its  vertex.  The  reason  of  so  small  a 
number  of  examples  being  worked  may  be,  that  it  is  not  easy  to  eliminate 
the  co-ordinates  of  the  fixed  and  rolling  curves  from  his  equations. 

The  case  in  which  one  curve  roUing  on  another  produces  a  circle  is  treated 
of  in  Willis's  Principles  of  Mechanism.     Class  C.     Boiling  Contact. 


6  THE    THEORY    OP    ROLLHiTO    CURVES. 

He  employs  the  same  method  of  finding  the  one  curve  from  the  other 
which  is  used  here,  and  he  attributes  it  to  Euler  (see  the  Acta  Petropolitana, 
Vol.  v.). 

Thus,  nearly  all  the  simple  cases  have  been  treated  of  by  different  authors; 
but  the  subject  is  still  far  from  being  exhausted,  for  the  equations  have  been 
applied  to  very  few  curves,  and  we  may  easily  obtain  new  and  elegant  proper- 
ties from  any  curve  we  please. 

Almost  all  the  more  notable  curves  may  be  thus  linked  together  in  a  great 
variety  of  ways,  so  that  there  are  scarcely  two  curves,  however  dissimilar, 
between  which  we  cannot  form  a  chain  of  connected  curves. 

This  will  appear  in  the  list  of  examples  given  at  the  end  of  this  paper. 


Let  there  be  a  curve  KAS,  whose  pole  is  at  C. 


THE  THEORY  OF  ROLLING  CURVES.  7 

Let  the  angle  DCA  =  6,  and   CA=r,  and  let 

Let  this  curve  remain  fixed  to  the  paper. 

Let  there  be  another  curve  BAT,  whose  pole  is  B. 

Let  the  angle  MBA  =  0t,  and  BA=r^,  and  let 

Let  this  curve  roll  along  the  curve  KAS  without  slipping. 
Then  the  pole  B  will  describe  a  third  curve,  whose  pole  is  C. 
Let  the  angle  DCB  =  0^,  and   CB  =  r„  and  let 

We  have  here  six  unknown  quantities  0,dAr,r^r^;  but  we  have  only  three 
equations  given  to  connect  them,  therefore  the  other  three  must  be  sought  for 
in  the  enunciation. 

But  before  proceeding  to  the  investigation  of  these  three  equations,  we  must 
premise  that  the  three  curves  will  be  denominated  as  follows  : — 
The  Fixed  Curve,  Equation,     e^  =  ^^{r^. 
The  Rolled  Curve,  Equation,   0.  =  <f>,{r,). 
Tlie  Traced  Curve,  Equation,   6^  =  4>.,{r^. 

When  it  is  more  convenient  to  make  use  of  equations  between  rectangular 
co-ordinates,  we  shall  use  the  letters  x^^,  x^^,  x^ij^.  We  shall  always  employ  the 
letters  s^s^^  to  denote  the  length  of  the  curve  from  the  pole,  p.p^p^  for  the  per- 
pendiculars from  the  pole  on  the  tangent,  and  q^q/i^  for  the  intercepted  part  of 
the  tangent. 

Between  these  quantities,  we  have  the  following  equations: — 

r  =  ^/^T?,  ^  =  tan-|, 

a?  =  r  cos  ^,  y  =  r  sin  6, 

r"  ydx  —  xdy 


jm'S         ""^w+w' 


THE    THEORY    OF     ROLLING    CURVES. 

rdr 

dS  _    xdx  +  ydy 


2=-r=7x!fi'  r- 


J{dxy  +  (dyY' 


'  "^    W      '^d^  daf 

We  come  now  to  consider  the  three  equations  of  rolling  which  are  involved 
in  the  enunciation.  Since  the  second  curve  rolls  upon  the  first  without  slipping, 
the  length  of  the  fixed  curve  at  the  point  of  contact  is  the  measure  of  the 
length  of  the  rolled  curve,  therefore  we  have  the  following  equation  to  connect 
the  fixed  curve  and  the  rolled  curve — 

«!  =  Sj. 

Now,  by  combining  this  equation  with  the  two  equations 

it  is  evident  that  from  any  of  the  four  quantities  6{r^6^r^  or  x^^x^^,  we  can 
obtain  the  other  three,  therefore  we  may  consider  these  quantities  as  known 
functions  of  each  other. 

Since  the  curve  rolls  on  the  fixed  curve,  they  must  have  a  common  tangent. 

Let  PA  be  this  tangent,  draw  BP,  CQ  perpendicular  to  PA,  produce  CQ, 
and  draw  BR  perpendicular  to  it,  then  we  have  CA=r^,  BA  =  r^,  and  CB  =  r,; 
CQ=p„  PB=p,,  and  BN=p,;   AQ  =  q„  AP  =  q„  and  CN=q,. 

Also  r,'=CR=CR  +  RR  =  (CQ  +  PBY+(AP-AQf 

=p,'  +  2p,p,  +p,'  +  r,'  -p,'  -  2q,q,  +  r,"  -p,' 
fz  =  n'  +  n'  +  2piPa  -  2q,q^. 
Since  the  first  curve  is  fixed  to  the  paper,  we  may  find  the  angle  6,. 
Thus  e,  =  DCB  =  DCA  +  ACQ  +  RCB 

=  e?.  +  tan-|  +  tan-|§ 
^,  =  ^,  +  tan--^  +  tan-^  ^^^^ 


TjdO^  Pi  +pi 


THE    THEORY    OF    ROLLING    CURVES.  » 

Thus  we  have  found  three  independent  equations,  which,  together  with  the 
equations  of  the  curves,  make  up  six  equations,  of  which  each  may  be  deduced 
from  the  others.  There  is  an  equation  connecting  the  radii  of  curvature  of  the 
three  curves  which  is  sometimes  of  use. 

The  angle  through  which  the  rolled  curve  revolves  during  the  description  of 
the  element  ds„  is  equal  to  the  angle  of  contact  of  the  fixed  curve  and  the 
rolling  curve,  or  to  the  sum  of  their  curvatures, 

ds^     ds^     ds. 

But  the  radius  of  the  rolled  curve  has  revolved  In  the  opposite  direction 
through  an  angle  equal  to  dO,,  therefore  the  angle  between  two  successive  posi- 
tions  of  r,   is   equal   to    -^-dd,.     Now    this    angle    is    the    angle    between    two 

successive  positions  of  the  normal  to  the  traced  curve,  therefore,  if  0  be  the 
centre  of  curvature  of  the  traced  curve,  it  is  the  angle  which  ds^  or  ds^  subtends 
at  0.     Let  OA^T,  then 

ds^     r4d^     ds,      ,^  _  ds^     ds,      ,. 

^J__J_      1    _^ 
•*•  '^'ds,   T~  R,     R,     ds/ 

-tAt^tJ    RJR.' 

As  an  example  of  the  use  of  this  equation,  we  may  examine  a  property 
of  the  logarithmic  spiral. 

In    this    curve,    p  =  mr,   and    R  =  —  ,   therefore    if    the   rolled    curve   be   the 
■^  m 

logarithmic  spiral 

/I        1\       1  ^m 

"^[t^tJ-r^v/ 

m_  1 

t~r:,* 

AO 
therefore  ^0  in  the  figure  =  ?ni2i,  and  -^  =  m. 

Let  the  locus  of  0,  or  the  evolute  of  the  traced  curve  LYBH,  be  the 
curve  OZY,  and  let  the  evolute  of  the  fixed  curve  KZAS  be  FEZ,  and  let 
us  consider  FEZ  as  the  fixed  curve,  and  OZF  as  the  traced  curve. 


10  THE    THEORY    OF    ROLLING    CURVES. 

Then  in  the  triangles  BPA,  AOF,  we  have  OAF=PBA,  and  ^='^  =  ^y 

therefore  the  triangles  are  similar,  and  FOA  =  APB  =  - ,  therefore  OF  is  perpen- 
dicular to  OA,  the  tangent  to  the  curve  OZY,  therefore  OF  is  the  radius  of 
the  curve  which  when  roUed  on  FEZ  traces  OZY,  and  the  angle  which  the 
curve  makes  with  this  radius  is  OFA=PAB  =  %mr^m,  which  is  constant,  there- 
fore the  curve,  which,  when  rolled  on  FEZ,  traces  OZY,  is  the  logarithmic 
spiral.  Thus  we  have  proved  the  following  proposition :  "  The  involute  of  the 
curve  traced  by  the  pole  of  a  logarithmic  spiral  which  rolls  upon  any  curve, 
is  the  curve  traced  by  the  pole  of  the  same  logarithmic  spiral  when  rolled  on 
the  involute  of  the  primary  curve." 

It  follows  from  this,  that  if  we  roll  on  any  curve  a  curve  having  the 
property  _2:»i  — Wjri,  and  roll  another  curve  having  Pi  =  'm^r^  on  the  curve  traced, 
and  so  on,  it  is  immaterial  in  what  order  we  roll  these  curves.  Thus,  if  we 
roll  a  logarithmic  spiral,  in  which  jp  =  mr,  on  the  nth  involute  of  a  circle  whose 
radius  is  a,  the  curve  traced  is  the  w+lth  involute  of  a  circle  whose  radius 
is  Jl-m\ 

Or,  if  we  roll  successively  m  logarithmic  spirals,  the  resulting  curve  is  the 
n  +  mth  involute  of  a  circle,  whose  radius  is 

aJl—m^  sll- m/,    Jkc. 

We  now  proceed  to  the  cases  in  which  the  solution  of  the  problem  may 
be  simplified.  This  simplification  is  generally  effected  by  the  consideration  that 
the  radius  vector  of  the  rolled  curve  is  the  normal  drawn  from  the  traced 
curve  to  the  fixed  curve. 

In  the  case  in  which  the  curve  is  rolled  on  a  straight  line,  the  perpen- 
dicular on  the  tangent  of  the  rolled  curve  is  the  distance  of  the  tracing  point 
from  the  straight  line ;  therefore,  if  the  traced  curve  be  defined  by  an  equation 
in  iCg  and  y„ 

'^.°p.=  /  "'„... (1)' 


and 


'••=^'^©^ ^'^- 


THE  THEORY  OF  ROLLING  CURVES.  11 

By   substituting   for   r,   in   the   first   equation,    its  value,  as  derived  from  the 
second,  we  obtain 

-■©■[©■-]=©'■ 

If    we    know    the    equation    to    the     rolled    curve,    we    may   find    (-7-^')     in 

terms   of   r,,    then   by   substituting   for   r,   its   value    in   the   second   equation,    we 

dx  (1 X 

have   an   equation   containing  x^   and   -^,    from   which  we   find   the  value  of  -t— ' 

dy,  du, 

in  terms  of  x^;  the  integration  of  this  gives  the  equation  of  the  traced  curve. 

As   an  example,    we  may  find  the  curve  traced  by  the  pole  of  a  hyperbolic 
spiral  which  rolls  on  a  straight  line. 

a 

fdrA'  _  rl 
,ddj  ~  a' 


The  equation  of  the  rolled  curve  is  6^  = 


-  •■©■-■[(IJ-]' 


dx^  _       ^3 
'*  dy,~Ja'-x,'' 

This   is   the   differential  equation  of  the  tractory  of  the  straight  line,   which 
is  the  curve  traced  by  the  pole  of  the  hyperbolic  spiral. 
By  eliminating  x^  in  the  two  equations,  we  obtain 

dr^_      /dxA 

This  equation  serves  to  determine  the  rolled  curve  when  the  traced  cuive 
is  given. 

As  an  example  we  shall  find  the  curve,  which  being  rolled  on  a  straight 
line,  traces  a  common  catenary. 

Let  the  equation  to  the  catenary  be 


'l(e'  +  e-^. 


12  THE  THEORY  OF  ROLLING  CURVES. 


Then 


dy,~N  a'       ' 


dr 


then  by  integration  ^  =cos'^  ( 1  j 


2a 
r  = 


1+COS0' 


This  is  the  polar  equation  of  the  parabola,  the  focus  being  the  pole ;  there- 
fore, if  we  roll  a  parabola  on  a  straight  line,  its  focus  will  trace  a  catenary. 

The  rectangiilar  equation  of  this  parabola  is  af  =  Aay,  and  we  shall  now 
consider  what  curve  must  be  rolled  along  the  axis  of  y  to  trace  the  parabola. 

By  the  second  equation  (2), 

n  =  ^9    /-4-  +  l>    but  x^^Pi, 
V  ^» 

.-.  r/=^/  +  4a", 
.-.  2a  =  Vr/-jp/  =  g'„ 
but  q^  is   the  perpendicular   on  the   normal,   therefore   the  normal   to  the  curve 
always  touches  a  circle  whose  radius  is  2a,  therefore  the  curve  is  the  involute 
of  this  circle. 

Therefore  we  have  the  following  method  of  describing  a  catenary  by  con- 
tinued motion. 

Describe  a  circle  whose  radius  is  twice  the  parameter  of  the  catenary;  roll  a 
straight  line  on  this  circle,  then  any  point  in  the  line  will  describe  an  involute 


THE    THEORY    OF    ROLLING     CURVES.  13 

of  the  circle  ;  roll  this  curve  on  a  straight  line,  and  the  centre  of  the  circle  will 
describe  a  parabola  ;  roll  this  parabola  on  a  straight  line,  and  its  focus  will  trace 
the  catenary  required. 

We  come  now  to  the  case  in  which  a  straight  line  rolls  on  a  curve. 

When  the  tracing-point  is  in  the  straight  line,  the  problem  becomes  that 
of  involutes  and  evolutes,  which  we  need  not  enter  upon ;  and  when  the  tracmg- 
point  is  not  in  the  straight  line,  the  calculation  is  somewhat  complex;  we  shall 
therefore  consider  only  the  relations  between  the  curves  described  in  the  first 
and  second  cases. 

Definition. — The  curve  which  cuts  at  a  given  angle  all  the  circles  of  a 
given  radius  whose  centres  are  in  a  given  curve,  is  called  a  tractory  of  the 
given  curve. 

Let  a  straight  line  roll  on  a  curve  A,  and  let  a  point  in  the  straight 
line  describe  a  curve  B,  and  let  another  point,  whose  distance  from  the  first 
point  is  b,  and  from  the  straight  line  a,  describe  a  curve  C,  then  it  is  evident 
that  the  curve  B  cuts  the  circle  whose   centre   is   in    C,   and  whose  radius  is  b, 

at  an  angle  whose  sine  is  equal  to   r,   therefore  the   curve   5   is   a   tractory   of 

the  curve  C. 

When  a  =  b,  the  curve  B  is  the  orthogonal  tractory  of  the  curve  C.  If 
tangents  equal  to  a  be  drawn  to  the  curve  B,  they  will  be  terminated  in 
the  curve  C;  and  if  one  end  of  a  thread  be  carried  along  the  curve  C,  the 
other  end  will  trace  the  curve  B. 

When  a  =  0,  the  curves  B  and  C  are  both  involutes  of  the  curve  A, 
they  are  always  equidistant  from  each  other,  and  if  a  circle,  whose  radius  is 
6,  be  rolled  on  the  one,  its  centre  will  trace  the  other. 

If  the  curve  A  is  such  that,  if  the  distance  between  two  points  measured 
along  the  curve  is  equal  to  6,  the  two  points  are  similarly  situate,  then  the 
curve  B  is  the  same  with  the  curve  C.  Thus,  the  curve  A  may  be  a  re- 
entrant curve,  the  circumference  of  which  is  equal  to  6. 

When  the  curve  -4  is  a  circle,  the  curves  B  and   C  are  always  the  same. 

The  equations  between  the  radii  of  curvature  become 

1     1  _  r 


14  THE    THEORY    OF    ROLLING    CURVES. 

When  a  =  0,  T=0,  or  the  centre  of  curvature  of  the  curve  B  is  at  the 
point  of  contact.  Now,  the  normal  to  the  curve  C  passes  through  this  point, 
therefore — 

"The  normal  to  any  curve  passes  through  the  centre  of  curvature  of  its 
tractory," 

In  the  next  case,  one  curve,  by  rolling  on  another,  produces  a  straight 
line.  Let  this  straight  line  be  the  axis  of  y,  then,  since  the  radius  of  the 
rolled  curve  is  perpendicular  to  it,  and  terminates  in  the  fixed  curve,  and 
since  these  curves  have  a  common  tangent,  we  have  this  equation, 

If  the  equation  of  the  rolled  curve  be  given,  find  -j-^  in  terms  of  r^,  sub- 
stitute Xi  for  r^,  and  multiply  by  x^,  equate  the  result  to  -^ ,  and  integrate. 

Thus,  if  the  equation  of  the  rolled  curve  be 

d  =  Ar-""  +  &c.  +  Kr-^  +  Lr'^  +  if  log  r  +  iVr  +  &c.  +  Zr"", 

^  =  -  n^r-(»+^)  -  &c.  -  2Kr-'  -  I/p-'  +  Mr''  +  N+  &c.  +  wZr"-^ 
dr 

-r-=  -  nAx~'*  -  &c.  -  2Kx~"-  -  Lx~^  +  M+  Nx  +  &c.  +  nZx", 
ax 

y  =  -^  Aa^-""  +  &c.  +  2Kx-'  -L\ogx  +  Mx  +  ^Naf  +  &c.  +  -^  Zx""^', 

which  is  the  equation  of  the  fixed  curve. 

If  the  equation  of  the  fixed  curve  be  given,  find  -^  in  terms  of  cc,  sub- 
stitute r  for  X,  and  divide  by  r,  equate  the  result  to  -t-,  and  integrate. 

Thus,  if  the  fixed  curve  be  the  orthogonal  tractory  of  the  straight  line, 
whose  equation  is 


y  =  a  log .  +  Ja^ 

a  +  \la^  —  x^ 

dy  _  Jo'  —  af 
dx~       X 


THE  THEORY  OF  ROLUNG  CURVES. 


15 


de  _Ja?-7* 
dr  r* 

0  =  cos"^ 


this  is  the  equation  to  the  orthogonal  tractory  of  a  circle  whose  diameter  is 
equal  to  the  constant  tangent  of  the  fixed  curve,  and  its  constant  tangent 
equal  to  half  that  of  the  fixed  curve. 

This  property  of  the  tractory  of  the  circle  may  be  proved  geometrically, 
thus — Let  P  be  the  centre  of  a  circle  whose  radius  is  PD,  and  let  CD  be 
a  line  constantly  equal  to  the  radius.  Let  BCP  be  the  curve  described  by 
the  point  C  when  the  point  D  is  moved  along  the  circumference  of  the  circle, 
then  if  tangents  equal  to  CD  be  drawn  to  the  curve,  their  extremities  will 
be  in  the  circle.  Let  ACH  be  the  curve  on  which  BCP  rolls,  and  let  OPE 
be  the  straight  line  traced  by  the  pole,  let  CDE  be  the  common  tangent, 
let  it  cut  the  circle  in  D,  and  the  straight  line  in  E. 


Then  CD  =  PD,  .'.  LDCP^  LDPC,  and  CP  is  perpendicular  to  OE, 
.'.  L  CPE=  LDCP+  LDEP.  Take  away  LDCP-^  L  DPC,  and  there  remains 
DPE=DEP,  .-.  PD=^DE,  .-.  CE=2PD. 


16  THE    THEORY    OF    ROLLING    CURVES. 

Therefore  the  curve  ACH  haa  a  constant  tangent  equal  to  the  diameter  of 
the  circle,  therefore  ACH  is  the  orthogonal  tractorj  of  the  straight  line,  which 
is  the  tractrix  or  equitangential  curve. 

The  operation  of  finding  the  fixed  curve  from  the  rolled  curve  is  what 
Sir  John  Leslie  calls  "  divesting  a  curve  of  its  radiated  structure." 

The  method  of  finding  the  curve  which  must  be  rolled  on  a  circle  to 
trace  a  given  curve  is  mentioned  here  because  it  generally  leads  to  a  double 
result,  for  the  normal  to  the  traced  curve  cuts  the  circle  in  two  points,  either 
of  which  may  be  a  point  in  the  rolled  curve. 

Thus,  if  the  traced  curve  be  the  involute  of  a  circle  concentric  with  the 
given  circle,  the  rolled  curve  is  one  of  two  similar  logarithmic  spirals. 

If  the  curve  traced  be  the  spiral  of  Archimedes,  the  rolled  curve  may  be 
either  the  hyperbolic  spiral  or  the  straight  line. 

In  the  next  case,  one  curve  rolls  on  another  and  traces  a  circle. 

Since  the  curve  traced  is  a  circle,  the  distance  between  the  poles  of  the 
fixed  curve  and  the  rolled  curve  is  always  the  same;  therefore,  if  we  fix  the 
rolled  curve  and  roll  the  fixed  curve,  the  curve  traced  will  still  be  a  circle, 
and,  if  we  fix  the  poles  of  both  the  curves,  we  may  roU  them  on  each  other 
without  friction. 

Let  a  be  the  radius  of  the  traced  circle,  then  the  sum  or  difference  of 
the  radii  of  the  other  curves  is  equal  to  a,  and  the  angles  which  they  make 
with  the  radius  at  the  point  of  contact  are  equal, 

.♦.  n-=±(a±r,)andn^^  =  r,^\ 

dO, _  ±(a±r^  dS, 
drt~        r,         dvi' 

If  we  know  the  equation  between  ^j  and  r,,  we  may  find  ^—  in  terms  of  r„ 

substitute  ±  (a  ±  r,)  for  r„  multiply  by       ^         \  and  integrate. 

Thus,  if  the  equation  between  6^  and  r^  be 

r,  =  a  sec  $,, 


TEU:    THEORY    OF    ROLLING   CURVES.  17 

which  is  the  polar  equation   of  a   straight   line  touching  the  traced  circle   whose 
equation  is  r  =  ay  then 

dd  _         a 

dr,  ~  r,  -Jr.'-a' 
a 


{r,±a)Jr,'±2r,a 
dO^     r^±a  a 


dr,        r,     (r,±a)  Jrf±2r^ 


a 


_    2a     _     2a 

Now,   since   the  rolling    curve   is   a   straight  line,   and   the   tracing   point   is 

not  in  its   direction,   we  may  apply    to   this    example    the    observations    which 

have  been  made  upon  tractories. 

2a 
Let,  therefore,   the  curve  ^  =  ^ — 7  be  denoted  by  A,  its  involute  by  B,  and 

the   circle  traced  by    C,   then   B  is  the  tractory   of   C;    therefore   the   involute 

2a 
of  the   curve   ^  =  ^ — r  is   the   tractory   of    the   circle,    the   equation   of  which   is 


^  =  cos"' /—  —  I.     The   curve  whose  equation  is  ^'=s — ;    seems  to  be  among 

spirals  what  the  catenary  is  among  curves  whose  equations  are  between  rec- 
tangular co-ordinates ;  for,  if  we  represent  the  vertical  direction  by  the  radius 
vector,  the  tangent  of  the  angle  which  the  curve  makes  with  this  line  is 
proportional  to  the  length  of  the  curve  reckoned  from  the  origin ;  the  point 
at  the  distance  a  from  a  straight  line  rolled  on  this  curve  generates  a  circle, 
and  when  rolled  on  the  catenary  produces  a  straight  line ;  the  involute  of  this 
curve  m  the  tractory  of  the  circle,  and  that  of  the  catenary  is  the  tractory 
of  the  straight  line,  and  the  tractory  of  the  circle  rolled  on  that  of  the  straight 
line  traces  the  straight  line ;  if  this  curve  is  rolled  on  the  catenary,  it  produces 
the  straight  line  touching  the    catenary    at    its    vertex ;    the   method   of  drawing 


18  THE    THEORY    OF    ROLLING    CURVES. 

tangents  is   the   same  as  in  the  catenary,  namely,  by  describing  a  circle 
radius  is  a  on  the  production  of  the  radius  vector,  and  drawing  a  tangent  to  the 
circle  from  the  given  point. 

In  the  next  case  the  rolled  curve  is  the  same  as  the  fixed  curve.  It  is 
evident  that  the  traced  curve  wiU  be  similar  to  the  locus  of  the  intersection 
of  the  tangent  with  the  perpendicular  from  the  pole ;  the  magnitude,  however, 
of  the  traced  curve  will  be  double  that  of  the  other  curve;  therefore,  if  we 
call  n  =  <^o^o  the  equation  to  the  fixed  curve,  r,  =  <f>,6,  that  of  the  traced  curve, 
we  have 

also,  £^  =  f. 

SimUarly,  r,  =  2p,  =  2r,f  =  A^  Ar,  (^J,   0,^6,-2  cos-  ^  . 

Similarly,  r„  =  2p„.,  =  2r„_,  ^  &c.  =  2^  (^^J  , 
and  ^^f. 

^„  =  ^„-7lC0S-f-\ 

'o 

V 
0n  =  6.  —  ncos~^  -^ . 

Let  e,  become  6^';   0„  6,'  and  ^ ,  ^.     Let  ^„^-^„  =  a, 

^„^  =  ^;-ncos-  ^, 
» «. 


a  =  ^„^-  e„  =  ^.^-^o-ncos-^  ^'  +n  cos-^  ^ 

-1  Pn  -1  Pn  O-    ,    ^0  ~  ^0 

\  cos  ^  ^^-^  —  COS  *  -^—  =  -  4 . 


THE  THEORY  OF  ROLLING  CURVES.  19 

Now,  cos"^  —   is   the  complement  of  the  angle  at  which  the  curve  cuts  the 

'  n 

radius  vector,  and  cos"'  — —cos"'  -^  is  the  variation  of  this  angle  when  6^  varies 
by  an  angle  equal  to  a.     Let  this  variation  =  (^  ;  then  if  6^  —  6 J  =  fi, 

^     n     n 
Now,  if  n  increases,  <f>  will  diminish ;   and  if  n  becomes  infinite, 

<^  =  ^  +  ^  =  0  when  a  and  )8  are  finite. 

Therefore,  when  n  is  infinite,  <}>  vanishes ;  therefore  the  curve  cuts  the  radius 
vector  at  a  constant  angle  ;  therefore  the  curve  is  the  logarithmic  spiral. 

Therefore,  if  any  curve  be  rolled  on  itself,  and  the  operation  repeated  an 
infinite  number  of  times,  the  resulting  curve  is  the  logarithmic  spiral 

Hence  we  may  find,  analytically,  the  curve  which,  being  rolled  on  itself, 
traces  itself. 

For  the  curve  which  has  this  property,  if  rolled  on  itself,  and  the  operation 
repeated  an  infinite  number  of  times,  will  still  trace  itself. 

But,  by  this  proposition,  the  resulting  curve  is  the  logarithmic  spiral ; 
therefore  the  curve  required  is  the  logarithmic  spiral.  As  an  example  of  a  curve 
rolling  on  itself,  we  will  take  the  curve  whose  equation  is 


n=2"a(cos|)". 
-1=2".  (sing  (oosf-; 
2"a'(cos^")'" 


.'.  r^  =  2p,=  2 


r,  =  2 


^2-a'(cosg%2-a^(sing    (cosg"^'^ 

2"a  cos  —  /       n\  „+i 

^^cos-j+(sm-j 


20  THE    THEORY    OF    ROLLING    CURVES. 

Now  ^1-^0= -cos-^^"= -cos-' cos -"  =  -^, 

"        n+1 
substituting  this  value  of  6^  in  the  expression  for  r^, 

r.  =  2-'a^cos--J     , 

similarly,  if  the  operation  be  repeated  ni  times,  the  resulting  curve  is 


*afcos— ^^y 
\      n  +  mj 


When  n=l,  the  curve  is 

r  =  2a  cos  9, 

the  equation  to  a  circle,  the  pole  being  in  the  circumference. 

When  n  =  2,  it  is  the  equation  to  the  cardioid 

r  =  4a  (cos -J  . 

In  order  to  obtain  the  cardioid  from  the  circle,  we  roll  the  circle  upon 
itself,  and  thus  obtain  it  by  one  operation ;  but  there  is  an  operation  which, 
bei6g  performed  on  a  circle,  and  again  on  the  resulting  curve,  will  produce  a 
cardioid,  and  the  intermediate  curve  between  the  circle  and  cardioid  is 


r  =  2 


>     /       20\i 


As  the  operation  of  rolling  a  curve  on  itself  is  represented  by  changing  n 
into  (n  +  1)  in  the  equation,  so  this  operation  may  be  represented  by  changing  n 
into  (w  +  i). 

Similarly  there  may  be  many  other  fractional  operations  performed  upon 
the  curves  comprehended  under  the  equation 

r  =  2"a(cos-j. 

We  may  also  find  the  curve,  which,  being  rolled  on  itself,  will  produce  a 
given  curve,  by  making  7i=  —  1. 


THE    THEORY    OF    ROLLING    CURVES.  21 

We  may  likewise  prove  by  the  same  method  as  before,  that  the  result  of 
performing  this  inverse  operation  an  infinite  number  of  times  is  the  logarithmic 
spiral. 

As  an  example  of  the  inverse  method,  let  the  traced  line  be  straight,  let 
its  equation  be 

r<,  =  2a  sec  d^, 
then  P^^p,^2a^2a_ 


therefore  suppressing  the  suflSx, 


=  ar, 


*  •  \d0j      a       ' 


dr 
r 


7i-'' 


■■&-') 


-      2a 
^~l-cos^' 

the  polar  equation  of  the  parabola  whose  parameter  is  4rt. 

The  last  case  which  we  shall  here  consider  affords  the  means  of  constructing 
two  wheels  whose  centres  are  fixed,  and  which  shall  roll  on  each  other,  so  that 
the  angle  described  by  the  first  shall  be  a  given  function  of  the  angle  described 
by  the  second. 

Let  0^  =  (f}0i,  then  r^  +  r^  =  a,  and  -j^  =  —  ; 

d0^        a-r^' 

Let  us  take  as  an  example,  the  pair  of  wheels  which  will  represent  the 
angular  motion  of  a  comet  in  a  parabola. 


THE    THEORY    OF    ROLLING    CURVES. 


Here  6^  =  tan  -^ , 


.  ^_ 


2  cos'  -^ 


a      2  +  cos  ^1 ' 

therefore  the  first  wheel  is  an  ellipse,  whose  major  axis  is  equal  to  |  of  the 
distance  between  the  centres  of  the  wheels,  and  in  which  the  distance  between 
the  foci  is  half  the  major  axis. 

Now  since  ^i  =  2  tan"'  B^  and  r^  =  a  -  r„ 

'•  1+    1 


a        ^2(2-^)' 

'-'-±;' 

a 
which  is  the  equation  to  the  wheel  which  revolves  with  constant  angular  velocity. 

Before  proceeding  to  give  a  list  of  examples  of  rolling  curves,  we  shall 
state  a  theorem  which  is  almost  self-evident  after  what  has  been  shewn  pre- 
viously. 

Let  there  be  three  curves.  A,  B,  and  C.  Let  the  curve  A,  when  rolled 
on  itself,  produce  the  curve  B,  and  when  rolled  on  a  straight  line  let  it 
produce  the  curve  C,  then,  if  the  dimensions  of  C  be  doubled,  and  B  be 
rolled  on  it,  it  will  trace  a  straight  line. 


A  Collection  of  Examples  of  Rolling  Curves. 

First.     Examples  of  a  curve  rolling  on  a  straight  line. 

Ex.  1.  When  the  rolling  curve  is  a  circle  whose  tracing-point  is  in  the 
circumference,  the  curve  traced  is  a  cycloid,  and  when  the  point  is  not  in  the 
circumference,  the  cycloid  becomes  a  trochoid. 

Ex.  2.  When  the  rolling  curve  is  the  involute  of  the  circle  whose  radius 
is  2a,  the  traced  curve  is  a  parabola  whose  parameter  is  4a. 


THE    THEORY    OF    ROLLING    CURVES.  23 

Ex.  3.  When  the  rolled  curve  is  the  parabola  whose  parameter  is  4a,  the 
traced  curv^e  is  a  catenary  whose  parameter  is  a,  and  whose  vertex  is  distant 
a  from  the  straight  line. 

Ex.  4.  "When  the  rolled  curve  is  a  logarithmic  spiral,  the  pole  traces  a 
straight  line  which  cuts  the  fixed  line  at  the  same  angle  as  the  spiral  cuts 
the  radius  vector. 

Ex.  5.  When  the  rolled  curve  is  the  hyperbolic  spiral,  the  traced  curve 
is  the  tractory  of  the  straight  line. 

Ex.  6.     When  the  rolled  curve  is  the  polar  catenary 


r       2a 


the   traced  curve   is   a  circle   whose   radius   is   a,  and  which  touches  the  straight 
line. 

Ex.  7.     When  the  equation  of  the  rolled  curve  is 

the  traced  curve  is  the  hyperbola  whose  equation  is 

y'  =  d'  +  a^. 

Second.     In   the   examples    of    a   straight  Hne   I'olling  on   a   curve,   we  shall 
use  the  letters  A^  B,  and  C  to  denote  the  three  curves  treated  of  in  page  22. 

Ex.  1.     When  the  curve  ^  is  a  circle  whose  radius  is  a,  then  the  cui-ve  B 
is  the  involute  of  that  circle,  and  the  curve  C  is  the  spiral  of  Archimedes,  r  =  ad. 

Ex.  2.     When  the  curve  ^   is  a  catenary  whose  equation  is 


the  curve  B  is  the  tractory  of  the  straight  line,  whose  equation  is 

X  I 

y  =  a  log ,  +  JcL'  —  -f^, 

a  +  V  a'  -  ar" 

and  C  is  a  straight  line  at  a  distance  a  from  the  vertex  of  the  catenary. 


24  THE  THEORY  OF  ROLLING  CURVES. 

Ex.  3.     When  tKe  curve  A  is  the  polar  catenaxy 
the  curve  B  is  the  tractory  of  the  circle 

and  the  curve  (7  is  a  circle  of  which  the  radius  is  - . 

Third.     Examples   of  one   curve  rolling   on  another,   and  tracing  a  straight 
line. 

Ex.  1.     The  curve  whose  equation  is 

0  =  Ar-"*  +  &c.  +  Kr-'  +  Lr'^  +  Jf  log  r  +  iVr  +  &c.  +  Zt^, 
when  rolled  on  the  curve  whose  equation  is 

n  —  1  71+  L 

traces  the  axis  of  y. 

Ex.  2.     The   circle   whose   equation   is  r  =  a  cos  ^  rolled  on  the  circle  whose 
radius  is  a  traces  a  diameter  of  the  circle. 

Ex.  3.     The  curve  whose  equation  is 


^=J'i- 


1  —  versm    - , 
a 


rolled  on  the  circle  whose  radius  is  a,  traces  the  tangent  to  the  circle. 

Ex.  4.  If  the  fixed  curve  be  a  parabola  whose  parameter  is  4a,  and  if  we 
roll  on  it  the  spiral  of  Archimedes  r  =  ad,  the  pole  will  trace  the  axis  of  the 
parabola. 

Ex.  5.  If  we  roll  an  equal  parabola  on  it,  the  focus  will  trace  the  directrix 
of  the  first  parabola. 

Ex.  6.  If  we  roll  on  it  the  curve  ^  =  t^  t^®  P^^®  "^^  ^^^^  ^^®  tangent 
at  the  vertex  of  the  parabola. 


THE  THEORY  OF  ROLLING  CURVES.  25 

Ex.  7.     If  we  roll  the  curve  whose  equation  is 

r  =  a  cos  (t^) 
on  the  ellipse  whose  equation  is 

the  pole  will  trace  the  axis  h. 

Ex.  8.     K  we  roll  the  curve  whose  equation  ia 

on   the  hyperbola  whose  equation  is 

the  pole  will  trace  the  axis  h. 

Ex,  9.     If  we  roll  the  lituus,  whose  equation  is 

on  the  hyperbola  whose  equation  is 

the  pole  will  trace  the  asymptote. 

Ex.  10.     The  cardioid  whose  equation  is 

r  =  a(H- cos  ^), 
rolled  on  the  cycloid  whose  equation  is 

12  =  a  versin"'  -  +  J2ax  -  ic*, 
^  a 

traces  the  base  of  the  cycloid. 

Ex.   11.     The  curve  whose  equation  is 

0  =  versm-'-  +  2^/ 1, 

rolled  on  the  cycloid,  traces  the  tangent  at  the  vertex. 


26  THE  THEORY  OF  ROLLING  CURVES. 

Ex.  12.    The  straight  line  whose  equation  is 

r  =  a  sec  B, 
rolled   on  a  catenary  whose   parameter  is  a,  traces  a  line  whose   distance  from 
the  vertex  is  a. 

Ex.  13.     The  part  of  the  polar  catenary  whose  equation  is 


rolled  on  the  catenary,  traces  the  tangent  at  the  vertex. 

Ex.  14.     The  other  part  of  the  polar  catenary  whose  equation  is 

rolled  on  the  catenary,  traces  a  line  whose  distance  from  the  vertex  is  equal  to  2a. 

Ex.  15.  The  tractory  of  the  circle  whose  diameter  is  a,  rolled  on  the 
tractory  of  the  straight  line  whose  constant  tangent  is  a,  produces  the  straight 
line. 

Ex.   16.     The  hyperbolic  spiral  whose  equation  is 

a 

'■=5' 

rolled  on  the  logarithmic  curve  whose  equation  is 

1      ^ 
2/  =  alog-, 

traces  the  axis  of  y  or  the  asymptote. 

Ex.  17.  The  involute  of  the  circle  whose  radius  is  a,  rolled  on  an  orthogonal 
trajectory  of  the  catenary  whose  equation  is 

traces  the  axis  of  y. 


Ex.   18.     The  curve  whose  equation  is 


THE  THEORY  OF  ROLLING  CURVES.  27 

rolled  on  the  witch,  whose  equation  is 

traces  the  asymptote. 

Ex.   19.     The  curve  whose  equation  is 

r  —  a  tan  Q, 
rolled  on  the  curve  whose  equation  is 

traces  the  axis  of  y. 

Ex.  20.     The  curve  whose  equation  is 

2r 


e= 


rolled  on  the  curve  whose  equation  is 

y  =  /  ,   or  r  =  a  tan  $, 

traces  the  axis  of  y. 

Ex.  21.    The  curve  whose  equation  is 

r  =  a  (sec  d  —  tan  0), 
rolled  on  the  curve  whose  equation  is 

2/  =  alogg+l), 
traces  the  axis  of  y. 

Fourth.     Examples  of  pairs  of  rolling  curves  which  have  their  poles  at  a  fixed 
distance  =  a. 


Ce  straight  line  whose  equation  is         ^=sec"'- 
..„ ,       . 

r 


2a 


The  polar  catenary  whose  equation  is     0=  ±fj  I  ± 

Ex.  2.     Two  equal  ellipses  or  hyperbolas  centered  at  the  foci. 
Ex.  3.     Two  equal  logarithmic  spirals. 

(Circle  whose  equation  is  r  =  2a  cos  6. 

Curve  whose  equation  is  ^-/J^  —  l  +  versin"^-. 


Ex.  4. 


28  THE  THEORY  OF  ROLLING  CURVES. 

fCaxdioid  whose  equation  is  r=2a(l+co8^). 


Ex.  5. 


Ex.  6. 


Ex.  7. 


[Curve  whose  equation  is  ^  =  sin"*-  +  log  ,— — —       . 

(Conchoid,  r  =  a  (secg- 1). 

Icurve,  ^  =  >A-? 

Spiral  of  Archimedes,  r  =  a0. 

T  T 

Curve,  ^  =  -  +  log 


+  sec"^  - 
a 


a       °  a 


f Hyperbolic  spiral,  r=-Q 

Ex.  8.     -! 


ICurve, 


a 


e'+l 


1 


Cpse  whose  equation  is  ^"^^2+ ~Q' 


Ex.  10. 


(Involute  of  circle,  ^~Ja^^^  ®®^"^  a ' 

'curve,  e^J^±2l±log(-±l+J^.±2'^. 


Fifth.     Examples  of  curves  rolling  on  themselves. 
Ex.  1.     When  the  curve  which  rolls  on  itself  is  a  circle,  equation 

r  =  a  cos  6, 
the  traced  curve  is  a  cardioid,  equation  r  =  a(l+cos^). 
Ex.  2.     When  it  is  the  curve  whose  equation  is 

r  =  2"a  (cos-j  , 
the  equation  of  the  traced  curve  is 

Ex.  3.    When  it  is  the  involute  of  the  circle,  the  traced  curve  is  the  spiral 
of  Archimedes. 


THE    THEORY    OF    ROLLING    CURVES.  29 

Ex.  4.    When  it  is  a  parabola,  the  focus  traces  the  directrix,  and  the  vertex 
traces  the  cissoid. 

Ex.  5.     When  it  is  the  hyperbolic  spiral,  the  traced  curve  is  the  tractory  of 
the  circle. 

Ex.  6.     When  it  is  the  polar  catenary,  the  equation  of  the  traced  curve  is 


J 


2a     ,  .   .,  r 

1  —  versin     -  . 

r  a 


Ex.  7.     When  it  is  the  curve  whose  equation  is 
the  equation  of  the  traced  curve  is  r  =  a  (e'  —  €~"). 


This  paper  commenced  with  an  outline  of  the  nature  and  history  of  the  problem  of  rolling 
curves,  and  it  was  shewn  that  the  subject  had  been  discussed  previously,  by  several  geometers, 
amongst  whom  were  De  la  Hire  and  Nicolfe  in  the  Memoir es  de  I'Academie,  Euler,  Professor 
Willis,  in  his  Principles  of  Mechanism,  and  the  Rev.  H.  Holditch  in  the  Cambridge  Philosophical 
Transactions. 

None  of  these  authors,  however,  except  the  two  last,  had  made  any  application  of  their 
methods ;  and  the  principal  object  of  the  present  communication  was  to  find  how  far  the  general 
equations  could  be  simplified  in  particular  cases,  and  to  apply  the  results  to  practice. 

Several  problems  were  then  worked  out,  of  which  some  were  applicable  to  the  generation 
of  curves,  and  some  to  wheelwork ;  while  others  were  interesting  as  shewing  the  relations  which 
exist  between  different  curves ;  and,  finally,  a  collection  of  examples  was  added,  as  an  illus- 
tration of  the  fertihty  of  the  methods  employed. 


[From  the  Transactions  of  the  Royal  Society  of  Edinburgh,  Vol.  XX.  Part  i,] 


III. — On  the  Equilibrium  of  Elastic  Solids. 


There  are  few  parts  of  mechanics  in  which  theory  has  differed  more  from 
experiment  than  in  the  theory  of  elastic  sohds. 

Mathematicians,  setting  out  from  very  plausible  assumptions  with  respect  to 
the  constitution  of  bodies,  and  the  laws  of  molecular  action,  came  to  conclusions 
which  were  shewn  to  be  erroneous  by  the  observations  of  experimental  philoso- 
phers. The  experiments  of  (Ersted  proved  to  be  at  variance  with  the  mathe- 
matical theories  of  Navier,  Poisson,  and  Lame  and  Clapeyron,  and  apparently 
deprived  this  practically  important  branch  of  mechanics  of  all  assistance  from 
mathematics. 

The  assumption  on  which  these  theories  were  founded  may  be  stated  thus : — 

Solid  bodies  are  composed  of  distinct  ^molecules,  which  are  kept  at  a  certain 
distance  from  each  other  by  the  opposing  principles  of  attraction  and  heat.  When 
the  distance  between  two  molecules  is  changed,  they  act  on  each  other  with  a  force 
whose  direction  is  in  the  line  joining  the  centres  of  the  molecules,  and  whose 
magnitude  is  equal  to  the  change  of  distance  multiplied  into  a  function  of  the 
distance  which  vanishes  when  that  distance  becomes  sensible. 

The  equations  of  elasticity  deduced  from  this  assumption  contain  only  one 
coefficient,  which  varies  with  the  nature  of  the  substance. 

The  insufficiency  of  one  coefficient  may  be  proved  from  the  existence  of 
bodies  of  different  degrees  of  solidity. 

No  effort  is  required  to  retain  a  liquid  in  any  form,  if  its  volume  remain 
unchanged;  but  when  the  form  of  a  solid  is  changed,  a  force  is  called  into 
action  which  tends  to  restore  its  former  figure ;    and  this  constitutes  the  differ- 


THE    EQUILIBRITJM    OF    ELASTIC    SOLIDS.  31 

ence  between  elastic  solids  and  fluids.  Both  tend  to  recover  their  vohirne,  but 
fluids  do  not  tend  to  recover  their  shape. 

Now,  since  there  are  in  nature  bodies  which  are  in  every  intermediate  state 
from  perfect  soHdity  to  perfect  liquidity,  these  two  elastic  powers  cannot  exist 
in  every  body  in  the  same  proportion,  and  therefore  all  theories  which  assign  to 
them  an  invariable  ratio  must  be  erroneous. 

I  have  therefore  substituted  for  the  assumption  of  Navier  the  following 
axioms  as  the  results  of  experiments. 

If  three  pressures  in  three  rectangular  axes  be  applied  at  a  point  in  an 
elastic  solid, — 

1.  TTie  sum  of  the  three  pressures  is  proportional  to  the  sum  of  the  com- 
pressions ichich  they  produce. 

2.  The  difference  between  two  of  the  pressures  is  propo7'tional  to  the  differ- 
ence of  the  compressions  which  they  produce. 

The  equations  deduced  from  these  axioms  contain  two  coefficients,  and  differ 
from  those  of  Navier  only  in  not  assuming  any  invariable  ratio  between  the 
cubical  and  linear  elasticity.  They  are  the  same  as  those  obtained  by  Professor 
Stokes  from  his  equations  of  fluid  motion,  and  they  agree  with  all  the  laws  of 
elasticity  which  have  been  deduced  from  experiments. 

In  this  paper  pressures  are  expressed  by  the  number  of  units  of  weight  to 
the  unit  of  surface ;  if  in  English  measure,  in  pounds  to  the  square  inch,  or 
in  atmospheres  of  15  pounds  to  the  square  inch. 

Compression  is  the  proportional  change  of  any  dimension  of  the  solid  caused 
by  pressure,  and  is  expressed  by  the  quotient  of  the  change  of  dimension  divided 
by  the  dimension  compressed'". 

Pressure  will  be  understood  to  include  tension,  and  compression  dilatation ; 
pressure  and  compression  being  reckoned  positive. 

Elasticity  is  the  force  which  opposes  pressure,  and  the  equations  of  elasticity 
are  those  which  express  the  relation  of  pressure  to  compression  f. 

Of  those  who  have  treated  of  elastic  solids,  some  have  confined  themselves 
to   the   investigation   of  the   laws   of  the  bending  and  twisting  of  rods,  without 

*  The  laws  of  pressure  and  compression  may  be  found  in  the  Memoir  of  Lam6  and  Clapeyrou.     St^t- 
note  A. 

t  See  note  B. 


32  THE    EQUIUBRIUM    OF    ELASTIC    SOLIDS. 

considering  the  relation  of  the  coefficients  which  occur  in  these  two  cases; 
while  others  have  treated  of  the  general  problem  of  a  solid  body  exposed  to 
any  forces. 

The  investigations  of  Leibnitz,  Bernoulli,  Euler,  Varignon,  Young,  La  Hire, 
and  Lagrange,  are  confined  to  the  equilibrium  of  bent  rods;  but  those  of 
Navier,  Poisson,  Lam^  and  Clapeyron,  Cauchy,  Stokes,  and  Wertheim,  are 
principally  directed  to  the  formation  and  application  of  the  general  equations. 

The  investigations  of  Navier  are  contained  in  the  seventh  volume  of  the 
Memoirs  of  the  Institute,  page  373;  and  in  the  AnnoUes  de  Chimie  et  de 
Physique,  2^  Sdrie,  xv.  264,  and  xxxviii.  435 ;  L'AppUcati(m  de  la  Micanique, 
Tom.  I. 

Those  of  Poisson  in  Mem.  de  I'lnstitut,  vm.  429 ;  Annales  de  Chimie,  2" 
S^rie,  XXXVI,  334 ;  xxxvii.  337 ;  xxxvtil  338 ;  xlu.  Journal  de  VEcole 
Polytechnique,  cahier  xx.,  with  an  abstract  in  Annales  de   Chimie  for  1829. 

The  memoir  of  MM.  Lam^  and  Clapeyron  is  contained  in  Crelle's  Mathe- 
matical Journal,  Vol.  vii. ;  and  some  observations  on  elasticity  are  to  be  found 
in  Lamp's   Cours  de  Physique, 

M.  Cauchy's  investigations  are  contained  in  his  Exercices  d! Analyse,  Vol.  in. 
p.    180,  published  in  1828. 

Instead  of  supposing  each  pressure  proportional  to  the  linear  compression 
which  it  produces,  he  supposes  it  to  consist  of  two  parts,  one  of  which  is  pro- 
portional to  the  linear  compression  in  the  direction  of  the  pressure,  while  the 
other  is  proportional  to  the  diminution  of  volume.  As  this  hypothesis  admits 
two  coefficients,  it  differs  from  that  of  this  paper  only  in  the  values  of  the 
coefficients  selected.     They  are   denoted   by  K  and  h,   and   K^fi  —  ^m,  k  =  m. 

The  theory  of  Professor  Stokes  is  contained  in  Vol.  vin.  Part  3,  of  the 
Cambridge   Philosophical   Transactions,   and   was   read   April    14,    1845. 

He  states  his  general  principles  thus : — "  The  capability  which  solids  possess 
of  being  put  into  a  state  of  isochronous  vibration,  shews  that  the  pressures 
called  into  action  by  small  displacements  depend  on  homogeneous  functions  of 
those  displacements  of  one  dimension.  I  shall  suppose,  moreover,  according  to 
the  general  principle  of  the  superposition  of  small  quantities,  that  the  pressures 
due  to  different  displacements  are  superimposed,  and,  consequently,  that  the 
pressures  are   linear  functions  of  the   displacements." 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  33 

Having  assumed  the  proportionality  of  pressure  to  compression,  he  proceeds 
to  define  his  coefficients.— "Let  -^8  be  the  pressures  corresponding  to  a  uniform 
linear  dilatation  8  when  the  solid  is  in  equilibrium,  and  suppose  that  it  becomes 
mA8,  in  consequence  of  the  heat  developed  when  the  solid  is  in  a  state  of  rapid 
vibration.  Suppose,  also,  that  a  displacement  of  shifting  parallel  to  the  plane 
xy,  for  which  8x  =  kx,  Sy=  -  hj,  and  hz  =  0,  calls  into  action  a  pressure  -  Bk 
on  a  plane  perpendicular  to  the  axis  of  x,  and  a  pressure  Bk  on  a  plane 
perpendicular  to  the  axis  of  y;  the  pressure  on  these  planes  being  equal  and 
of  contrary  signs;  that  on  a  plane  perpendicular  to  z  being  zero,  and  the  tan- 
gential   forces    on    those    planes   being   zero."      The   coefficients   A    and   B,   thus 

defined,  when  expressed  as  in  this  paper,  are  ^  =  3/x,,  B  =  -. 

Professor  Stokes  does  not  enter  into  the  solution  of  his  equations,  but  gives 
their  results  in  some  particular  cases. 

1.  A  body  exposed  to  a  uniform  pressure  on  its  whole  surface. 

2.  A  rod  extended  in  the  direction  of  its  length. 

3.  A  cylinder  twisted  by  a  statical  couple. 

He  then  points  out  the  method  of  finding  A  and  B  from  the  last  two  cases. 

While  explaining  why  the  equations  of  motion  of  the  luminiferous  ether  are 
the  same  as  those  of  incompressible  elastic  solids,  he  has  mentioned  the  property 
of  jylasticity  or  the  tendency  which  a  constrained  body  has  to  relieve  itself 
from  a  state  of  constraint,  by  its  molecules  assuming  new  positions  of  equi- 
librium. This  property  is  opposed  to  Hnear  elasticity  ;  and  these  two  properties 
exist  in  all  bodies,  but  in  variable  ratio. 

M.  Wertheim,  in  Annales  de  Chimie,  3«  Sdrie,  xxiii.,  has  given  the  results 
of  some  experiments  on  caoutchouc,  from  which  he  finds  that  K=k,  or  fi  =  ^m; 
and  concludes  that  k  =  K  in  all  substances.  In  his  equations,  fi  is  therefore 
made  equal  to  f  m. 

The  accounts  of  experimental  researches  on  the  values  of  the  coefficients 
are  so  numerous  that  I  can  mention  only  a  few. 

Canton,  Perkins,  (Ersted.  Aime,  CoUadon  and  Sturm,  and  Regnault,  have 
determined  the  cubical  compressibilities  of  substances;  Coulomb,  Duleau,  and 
Giulio,  have  calculated  the  linear  elasticity  from  the  torsion  of  wires;  and  a 
great  many  observations  have  been  made  on  the  elongation  and  bending  of  beams. 

VOL.  I.  ^ 


34  THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 

I  have  found  no  account  of  any  experiments  on  the  relation  between  the 
doubly  refracting  power  communicated  to  glass  and  other  elastic  solids  by  com- 
pression, and  the  pressure  which  produces  it^^" ;  but  the  phenomena  of  bent  glass 
seem  to  prove,  that,  in  homogeneous  singly-refracting  substances  exposed  to 
pressures,  the  principal  axes  of  pressure  coincide  with  the  principal  axes  of 
double  refraction ;  and  that  the  diflference  of  pressures  in  any  two  axes  is 
proportional  to  the  difference  of  the  velocities  of  the  oppositely  polarised  rays 
whose  directions  are  parallel  to  the  third  axis.  On  this  principle  I  have 
calculated  the  phenomena  seen  by  polarised  light  in  the  cases  where  the  solid 
is  bounded  by  parallel  planes. 

In  the  following  pages  I  have  endeavoured  to  apply  a  theory  identical 
with  that  of  Stokes  to  the  solution  of  problems  which  have  been  selected  on 
account  of  the  possibility  of  fulfilling  the  conditions.  I  have  not  attempted  to 
extend  the  theory  to  the  case  of  imperfectly  elastic  bodies,  or  to  the  laws  of 
permanent  bending  and  breaking.  The  solids  here  considered  are  supposed  not 
to  be  compressed  beyond  the  limits  of  perfect  elasticity. 

The  equations  employed  in  the  transformation  of  co-ordinates  may  be  found 
in  Gregory's  Solid  Geometry. 

I  have  denoted  the  displacements  by  Zx,  By,  Bz.  They  are  generally  denoted 
by  a,  /8,  y ;  but  as  I  had  employed  these  letters  to  denote  the  principal  axes 
at  any  point,  and  as  this  had  been  done  throughout  the  paper,  I  did  not  alter 
a  notation  which  to  me  appears  natural  and  intelligible. 


The  laws  of  elasticity  express  the  relation  between  the  changes  of  the 
dimensions  of  a  body  and  the  forces  which  produce  them. 

These  forces  are  called  Pressures,  and  their  effects  Compressions.  Pressures 
are  estimated  in  pounds  on  the  square  inch,  and  compressions  in  fractions  of  the 
dimensions  compressed. 

Let  the  position  of  material  points  in  space  be  expressed  by  their  co-ordinates 
X,  y,  and  z,  then  any  change  in  a  system  of  such  points  is  expressed  by  giving 
to   these   co-ordinates  the  variations  Bx,  By,  Bz,  these  variations  being  functions  of 

X,  y,  2. 

*  See  note  C. 


THE    EQUILIBRIUM    OF     ELASTIC    SOLIDS.  35 

The  quantities  Sx,  Sy,  8z,  represent  the  absolute  motion  of  each  point  in 
the  directions  of  the  three  co-ordinates ;  but  as  compression  depends  not  on 
absolute,  but  on  relative  displacement,  we  have  to  consider  only  the  nine 
quantities — 


dSx 

dSx 

dhx 

dx  ' 

dy' 

dz' 

dSy 
dx  ' 

dhy 
dy' 

dSij 

dz  ' 

dSz 
dx' 

dhz 

dy- 

dBz 

dz  ' 

Since  the  number  of  these  quantities  is  nine,  if  nine  other  independent 
quantities  of  the  same  kind  can  be  found,  the  one  set  may  be  found  in  terms 
of  the  other.     The  quantities  which  we  shall  assume  for  this  purpose  are— 

1.  Three   compressions,    — ,   —■ ,   — ,    in    the    directions   of   three   principal 

a       Id       y 

axes  a,  yS,  y. 

2.  The  nine  direction-codnes  of  these  axes,  with  the  six  connecting  equa- 
tions, leaving  three  independent  quantities.     (See  Gregory's  Solid  Geometry.) 

3.  The  small  angles  of  rotation  of  this  system  of  axes  about  the  axes  of 
x,  y,  z. 

The  cosines  of  the  angles  which  the  axes  of  x,  y,  z  make  with  those  of 
a,  ^,  y  are 

cos(aOa-)=aj,  cos  {^Ox)  =  \,  co%(yQ)x)  =  c,, 
cos  (aOy)  =  tto,  _cos  {fiOy)  =  h„,  cos  (yO^/)  =  c., 
cos  (aOz)  =a3,    cos  (/SOz)  =63,    cos  {yOz)  =  c,. 

These  direction-cosines  are  connected  by  the  six  equations, 
a^  +  h{  +  Ci'  =  1 ,  «i«s  +  ^h  +  CjC,  =  0, 

a./  -I-  h^  +  c,'  =  1 ,  a^a^  +  h.h^  +  cx^  =  0, 

a;  +  63'  +  Gj'  =  1 ,  a/t,  +  bj),  +  c^c,  =  0. 

The  rotation  of  the  system  of  axes  a,  13,  y,  round  the  axis  of 
x,  from  y  to  z,    =B0^, 
y,  from  z  to  x,  =S^j, 
z,  from  x  to  y,  =^0/, 


36 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 


By    resolving    the    displacements    8a,   h/S,  By,  B6„  B9.„  Z6„   in   the  directions 
of  the  axes  x,  y,  z,  the  displacements  in  these  axes  are  found  to  be 
hx  =  a,8a  +  h,Bp  +  c3y  -Be^  +  Bd,y, 
By  =  aM  +  h,Bl3  -f  c,By  -  Bd,x  +  Bd.z, 
Bz  =  a,Ba  +  hM  +  CsBy  -  BO^  +  Bd,x. 
Sa      .^     ^Si8 


But 


B^^rf,    and  8y  =  y^, 


and  Q.  =  a^x  +  a^  +  a.^,    /3  =  b,x  +  h^  +  h.^,    and    y  =  c,x  +  c,y -h  c^z. 

Substituting  these  values  of  Sa,  Sy8,  and  By  in  the  expressions  for  Bx,  By, 
Bz,  and  differentiating  with  respect  to  x,  y,  and  z,  in  each  equation,  we  obtain 
the  equations 

dBx     Ba,    ,.  8/8,2  ,  ^y 


dy      a  ^  y 

dBz  _  Ba 
dz 


a  p  y 


(1)- 


dBx      Ba  B^  T  J       By         ,5s/, 

dy       a     '        ^  y 


a 

dBx      Ba 
dz        a 

Ba 


BI3 

J' 

8^ 


dz        a  p  y 


dBy     Ba  BB  T  ^      By 

dx       a  p  y 


Be, 

c.f^  +  Bdi 

Be, 


-J—  =  —  ctjCti  +  -^  6361  +  -^  C3C1  +  8^2 


dZz 
dx 

dBz 


8^ 
a 

Sa 


8^ 
S/8 


r 


Be, 


Equations  of 
compression. 


{2). 


Equations  of  the  equilibnum  of  an  element  of  the  solid. 
The  forces  which  may  act  on  a  particle  of  the  solid  are : — 

1.  Three  attractions  in  the  direction  of  the  axes,  represented  by  X,   Y,  Z. 

2.  Six  pressures  on  the  six  faces. 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 


37 


3.     Two  tangential  actions  on  each  face. 

Let  the  six  faces   of  the  small  parallelopiped  be  denoted  by  x^,  3/,,  z„  x^  y„ 
and  z,,  then  the  forces  acting  on  x^  are : — 

1.  A  normal  pressure  jp,  acting  in  the  direction  of  x  on  the  area  dydz, 

2.  A   tangential   force   g,  acting  in  the  direction  of  y  on  the  same  area. 

3.  A   tangential  force   q^  acting  in   the   direction   of   z   on   the   same   area, 
and  so  on  for  the  other  five  faces,  thus : — 

Forces  which  act  in  the  direction  of  the  axes  of 

a;  2/  z 


On  the  face    a:, 

—  'p^dydz 

-  q^dydz 

-q.'dydz 

^. 

{P^'r  J^dx)dydz 

(^3 + 7^  ^^)  c?yc?x 

(q.'+-^^dx)dydz 

2/1 

—  q^dzdx 

—p^dzdx 

—  q.dzdx 

y-x 

{q\  +  ^dy)dzdx 

{p.+^dy)dzdx 

(q,  +  ^dy)dzdx 

Zi 

—  q^dxdy 

—  q^dxdy 

—p^dxdy 

^2 

fe+  -4^dz)dxdy 

(q^+^dz)dxdy 

(p.  +  ^dz)dxdy 

Attractions, 

pXdxdydz 

p  Ydxdydz 

pZdxdydz 

Taking  the  moments  of  these  forces  round  the  axes  of  the  particle,  we  find 

?i'  =  ?i,    q^=q.^   qz=qz', 

and   then  equating  the  forces  in  the   directions   of  the  three   axes,   and  dividing 
by  dx,  dy,  dz,  we  find  the  equations  of  pressures, 


dy      dz      dx     ^ 
dz      dx      dy      '^ 


Equations  of  Pressures. 


(3). 


38 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 


The  resistance  which  the  sohd  opposes  to  these  pressures  is  called  Elasticity, 
and  is  of  two  kinds,  for  it  opposes  either  change  of  volume  or  change  of  Jigure. 
These  two  kinds  of  elasticity  have  no  necessary  connection,  for  they  are  possessed 
in  very  different  ratios  by  different  substances.  Thus  jelly  has  a  cubical  elas- 
ticity little  different  from  that  of  water,  and  a  linear  elasticity  as  small  as  we 
please ;  while  cork,  whose  cubical  elasticity  is  very  small,  has  a  much  greater 
Imear  elasticity  than  jelly. 

Hooke  discovered  that  the  elastic  forces  are  proportional  to  the  changes 
that  excite  them,  or  as  he  expressed  it,  "  Ut  tensio  sic  vLs." 

To  fix  our  ideas,  let  us  suppose  the  compressed  body  to  be  a  parallelepiped, 
and  let  pressures  Pi,  Pj,  P3  act  on  its  faces  in  the  direction  of  the  axes 
a>   A  y,   which  will   become    the    principal   axes   of    compression,    and   the    com- 


pressions will  be 


So.      8^     Sy 
a'    ^'     y 


The  fundamental  assumption  from  which  the  following  equations  are  deduced 
is  an  extension  of  Hooke's  law,  and  consists  of  two  parts. 

I.  The  sum  of  the  compressions  is  proportional  to  the  sum  of  the  pressures. 

II.  The   difference   of  the   compressions   is   proportional   to  the   difference   of 
the  pressures. 


These  laws  are  expressed  by  the  following  equations 


I.    (P.  +  P,  +  P.)  =  3,(^  +  f +  ^ 


(4). 


II. 


(P,-P,)  =  m 


(P._p.)  =  „,g_^ 


(P.-P,)  =  m 


rv        ^rts  T    Equations  of  Elasticity. 

h 
7 

By      Ba 


(5). 


The  quantity   fj.   is  the   coefiicient  of  cubical  elasticity,  and  m  that  of  linear 
elasticity. 


THE    EQUILrBRIUM    OF    ELASTIC    SOLmS. 


39 


By   solving  these   equations,   the   values  of  the  pressures  P„  P,,  P„  and  the 

8a      8^     Sy  ,      r       J 

compressions  — '    ~S '  ^^7  ^^  found. 


a       \9/x      3m/  ^  ^      m 

!  =  (!_    M(p.  +  P,  +  p.)  +  lp, 

j3       \9/x      3  m/  ^    *  ^      ?7i     ' 

?r  =  (_L_  i\(P_+P_+P_)  +  ip_ 

y       \9/z      3m/  ^  ^      m 


(6). 


(7). 


From  these   values   of  the  pressures   in   the   axes   a,  )8,  y,  may   be   obtained.. 
the  equations  for  the  axes  x,  y,  z,  by  resolutions  of  pressures  and  compressions*. 


For 


and 


q  =  aaP^  +  hhP,  +  ccP, ; 
,       .    .  IdZx  ,  d%y  ,  d8z\  .      d8x' 

,       .    V  IdZx  .  d8y  ,  d8z\        dBy 

,       ,    ,  fdSx  ,  d8y  ,  rfSj\  ,      dSz 

m  /c?Sz      c?Sx 


(8)- 


2  Vo?a;      c?2 


.(9). 


See  the  Memoir  of  Lame  and  Clapeyron,  and  note  A. 


40 


THE    EQUIUBRIUM    OP    ELASTIC    SOLIDS. 


d$X       /I  1  \  ,       ,         ,       N   ,     1 


(10). 


dy         *     ax        '    m^ 
dz  dy  m  ^ 


d^ 
dx 


dz  m^ 


(11). 


By   substituting  in  Equations  (3)  the   values   of  the   forces  given  in   Equa- 
tions (8)  and  (9),  they  become 


(12). 


These   are  the   general   equations   of  elasticity,    and   are   identical  with  those 
of  M.    Cauchy,   in   his   Exercices  d' Analyse,  Vol.  ni.,  p.   180,   published   in    1828, 

where  h  stands  for  m,   and   K  for  ft  -  o"  >  and  those  of  Mr  Stokes,  given  in  the 

Cambridge   Philosophical    Transactions,    Vol.    viii.,   part   3,   and  numbered   (30); 

in  his  equations  ^  =  3/x,  B  =  —  . 

If   the    temperature   is    variable  from   one  part   to  another    of    the    elastic 

soHd,  the  compressions  -y- ,    -r^,   -J^ ,    at   any   point   will   be  diminished  by  a 

quantity   proportional  to  the  temperature  at  that  point.     This  prmciple  is  applied 
in  Cases  X.  and  XI.     Equations  (10)  then  become 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 


41 


dy 


^  =  fe  -  3mj  (P^-^P^+P^)  +  '^^^^P^ 


(13). 


CfV  being  the  linear  expansion  for  the  temperature  v. 

Having  found  the  general  equations  of  the  equilibrium  of  elastic  solids,  I 
proceed  to  work  some  examples  of  their  application,  which  afford  the  means  of 
determining  the  coefficients  /t,  m,  and  o),  and  of  calculating  the  stiffness  of 
solid  figures.  I  begin  with  those  cases  in  which  the  elastic  soHd  is  a  hollow 
cylinder  exposed  to  given  forces  on  the  two  concentric  cylindric  surfaces,  and 
the  two  parallel  terminating  planes. 

In  these   cases   the   co-ordinates   x,  y,   z  are   replaced   by   the   co-ordinates 
x  =  x,  measured  along  the  axis  of  the  cylinder. 
2/  =  r,  the  radius  of  any  point,  or  the  distance  from  the  axis. 
z  —  rd,  the  arc  of  a  circle  measured  from  a  fixed   plane   passing 
through  the  axis. 

Px  =  o,  are  the  compression  and  pressure  in  the  direction   of  the 
axis  at  any  point. 

-^  =  -J— ,  Pi  =p,  are  the  compression  and  pressure  in  the  direction  of  the 

radius. 

dBz      dhrd      Br  .  .  _  .      ,      ,.       .         -    1 

~dz~'db¥~l^'  JP8  =  ?,  are  the  compression  and  pressure  m  the  direction  of  the 

tangent. 

Equations  (9)  become,  when  expressed  in  terms  of  these  co-ordinates — 

m    doO 


dZx 
dx 


dSx 
dx 


m    dB0 

m    dSx 
dr 


*=2 


.(14). 


The  length  of  the  cylinder  is  h,  and  the  two  radii  a,  and  a,  in  every 
VOL.  I.  G 


42  THE    EQUIUBRnJM    OF    ELASTIC    SOLIDS. 


Case  I. 


The  first  equation  is  applicable  to  the  case  of  a  hollow  cylinder,  of  which 
the  outer  surface  is  fixed,  while  the  inner  surface  is  made  to  turn  through 
a  small  angle  Bd,  by  a  couple  whose  moment  is  M. 

The  twisting  force  M  is  resisted  only  by  the  elasticity  of  the  solid,  and 
therefore  the  whole  resistance,  in  every  concentric  cylindric  surface,  must  be  equal 
to  M. 

The  resistance  at  any  point,  multiplied  into  the  radius  at  which  it  acts,  is 
expressed  by 

m   „  dhd 

Therefore  for  the  whole  cylindric  surface 

ar 
Whence  8,=_^^  (1,_1.) , 

^^  "'  =  2^&-i) ('«>■ 

The  optical  effect  of  the  pressure  of  any  point  is  expressed  by 

I=<oq,b  =  <o.^^ (15). 

Therefore,  if  the  solid  be  viewed  by  polarized  light  (transmitted  parallel  to 
the  axis),  the  difference  of  retardation  of  the  oppositely  polarized  rays  at  any 
point  in  the  solid  will  be  inversely  proportional  to  the  square  of  the  distance  fi-om 
the  axis  of  the  cylinder,  and  the  planes  of  polarization  of  these  lays  will  be 
inclined  45"  to  the  radius  at  that  point. 

The  general  appearance  is  therefore  a  system  of  coloured  rings  arranged 
oppositely  to  the  rings  in  uniaxal  crystals,  the  tints  ascending  in  the  scale  as 
they  approach  the  centre,  and  the  distance  between  the  rings  decreasing  towards 
the  centre.  The  whole  system  is  crossed  by  two  dark  bands  inclined  45*  to  the 
plane  of  primitive  polarization,  when  the  plane  of  the  analysing  plate  is  perpen- 
dicular to  that  of  the  first  polarizing  plate. 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  43 

A  jelly  of  isinglass  poured  when  hot  between  two  concentric  cylinders  forms, 
when  cold,  a  convenient  solid  for  this  experiment ;  and  the  diameters  of  the  rings 
may  be  varied  at  pleasure  by  changing  the  force  of  torsion  appUed  to  the  interior 
cylinder. 

By  continuing  the  force  of  torsion  while  the  jeUy  is  allowed  to  dry,  a  hard 
plate  of  isinglass  is  obtained,  which  still  acts  in  the  same  way  on  polarized  light, 
even  when  the  force  of  torsion  is  removed. 

It  seems  that  this  action  cannot  be  accounted  for  by  supposing  the  interior 
parts  kept  in  a  state  of  constraint  by  the  exterior  parts,  as  in,  unannealed  and 
heated  gla^s ;  for  the  optical  properties  of  the  plate  of  isinglass  are  such  as 
would  indicate  a  strain  preserving  in  every  part  of  the  plate  the  direction  of 
the  original  strain,  so  that  the  strain  on  one  part  of  the  plate  cannot  be  main- 
tained by  an  opposite  strain  on  another  part. 

Two  other  uncrystallised  substances  have  the  power  of  retaining  the  polariz- 
ing structure  developed  by  compression.  The  first  is  a  mixture  of  wax  and  resin 
pressed  into  a  thin  plate  between  two  plates  of  glass,  as  described  by  Sir  David 
Brewster,  in  the  Philosophical  TransoLctions  for  1815  and  1830. 

When  a  compressed  plate  of  this  substance  is  examined  with  polarized  light, 
it  is  observed  to  have  no  action  on  light  at  a  perpendicular  incidence ;  but  when 
inclined,  it  shews  the  segments  of  coloured  rings.  This  property  does  not  belong 
to  the  plate  as  a  whole,  but  is  possessed  by  every  part  of  it.  It  is  therefore 
similar  to  a  plate  cut  from  a  uniaxal  crystal  perpendicular  to  the  axis. 

I  find  that  its  action  on  light  is  like  that  of  a  jpositive  crystal,  while  that 
of  a  plate  of  isinglass  similarly  treated  would  be  negative. 

The  other  substance  which  possesses  similar  properties  is  gutta  percha.  This 
substance  in  its  ordinary  state,  when  cold,  is  not  transparent  even  in  thin  films; 
but  if  a  thin  film  be  drawn  out  gradually,  it  may  be  extended  to  more  than 
double  its  length.  It  then  possesses  a  powerful  double  refraction,  which  it 
retains  so  strongly  that  it  has  been  used  for  polarizing  light""'.  As  one  of  its 
refractive  indices  is  nearly  the  same  as  that  of  Canada  balsam,  while  the  other 
is  very  different,  the  common  surface  of  the  gutta  percha  and  Canada  balsam 
will  transmit  one  set  of  rays  much  more  readdy  than  the  other,  so  that  a  film 
of  extended  gutta  percha  placed  between  two  layers   of  Canada   balsam  acts  like 

*  By  Dr  Wright,  I  believe. 


44  THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 

a  plate  of  nitre  treated  in  the  same  way.  That  these  films  are  in  a  state  of 
constraint  may  be  proved  by  heating  them  slightly,  when  they  recover  their 
original  dimensions. 

As  all  these  permanently  compressed  substances  have  passed  their  limit  of 
perfect  elasticity,  they  do  not  belong  to  the  class  of  elastic  solids  treated  of  in 
this  paper ;  and  as  I  cannot  explain  the  method  by  which  an  imcrystallised  body 
maintains  itself  in  a  state  of  constraint,  I  go  on  to  the  next  case  of  twisting, 
which  has  more  practical  importance  than  any  other.  This  is  the  case  of  a 
cylinder  fixed  at  one  end,  and  twisted  at  the  other  by  a  couple  whose  moment 
is  M. 

Case  II. 

In  this  case  let  hB  be  the  angle  of  torsion  at  any  point,  then  the  resistance 
to  torsion  in  any  circular  section  of  the  cylinder  is  equal  to  the  twisting  force  M, 

The  resistance  at  any  point  in  the  circular  section  is  given  by  the  second 
Equation  of  (14). 


?2  =  1^^ 


dx  ' 


This  force  acts  at  the  distance  r  from  the  axis ;  therefore  its  resistance  to  torsion 
will  be  q.r,  and  the  resistance  in  a  circular  annulus  will  be 

q^r^Ttrdr  =  mirr'  -r-  dr 

and  the  whole  resistance  for  the  hollow  cylinder  will  be  expressed  by 

„,    mn  dS6 ,    ^       ,.  /,^v 


720  M 


^(-1-] (17). 


In  this  equation,  m  is  the  coefl&cient  of  linear  elasticity;  a^  and  a^  are  the 
radii  of  the  exterior  and  interior  surfaces  of  the  hollow  cyUnder  in  inches ;  M  is 
the  moment  of  torsion  produced  by  a  weight  acting  on  a  lever,  and  is  expressed 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  45 

bj  the  product  of  the  number  of  pounds  in  the  weight  into  the  number  of  inches 
in  the  lever;  b  is  the  distance  of  two  points  on  the  cylinder  whose  angular 
motion  is  measured  by  means  of  indices,  or  more  accurately  by  small  mirrors 
attached  to  the  cylinder ;  n  is  the  difference  of  the  angle  of  rotation  of  the  two 
indices  in  degrees. 

This  is  the  most  accurate  method  for  the  determination  of  m  independently 
of  /x,  and  it  seems  to  answer  best  with  thick  cylinders  which  cannot  be  used 
with  the  balance  of  torsion,  as  the  oscillations  are  too  short,  and  produce  a 
vibration  of  the  whole  apparatus. 


Case  III. 

A  hollow  cylinder  exposed  to  normal  pressures  only.  When  the  pressures 
parallel  to  the  axis,  radius,  and  tangent  are  substituted  for  p^,  p^,  and  pt, 
Equations  (10)  become 

S  =  (i-34)(^+^-^^)  +  ^ (^«)- 

^^t^(±-±]io+p  +  q)  +  :^q (20). 

By  multiplying  Equation   (20)  by  r,   differentiating  with  respect  to  r,  and 

comparing  this  value  of  —j—  with  that  of  Equation  (19), 


p-q  _(J__  _1\  /^  .  ^  .  ^\  _  i  ^ 
rm  "  \9/x     3m/  \dr     dr     drj     m  dr ' 


The  equation  of  the  equilibrium  of  an  element  of  the  solid  is  obtained  by 
considering  the  forces  which  act  on  it  in  the  direction  of  the  radius.  By 
equating  the  forces  which  press  it  outwards  with  those  pressing  it  rnwarde,  we 
find  the  equation  of  the  equiHbrium  of  the  element, 

ir£  =  4 (21). 

r         dr 


46  THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 

By  comparing  this  equation  witli  the  last,  we  find 

\9fi     Zmj  dr     \9/i  ^  3m/  \dr  ^  drj 
Integrating, 

Since  o,  the  longitudinal  pressure,  is  supposed   constant,  we  may  assume 

c  -(^-^]o 
'      \9u,      3m/        .     ,    . 

c.  = 12  =(^  +  g)- 

9/x,      3  m 
Therefore  q—p  =  c^  —  2p,  therefore  by  (21), 

a  linear  equation,  which  gives 


1  ^c, 
^  =  ^3^  +  2- 

The  coefficients  Cj  and  Cj  must  be  found  from  the  conditions  of  the  surface 
of  the  soHd.  If  the  pressure  on  the  exterior  cylindric  surface  whose  radius  is  a, 
be  denoted  by  A,,  and  that  on  the  interior  surface  whose  radius  is  a^  by  A,, 

then  p  =  h^  when  r  =  ai 
and  p  =  h.j  when  r  =  a^ 
and  the  general  value  of  p  is 

_a^h^  —  a^\     a^a^  h^  —  h^  /22\ 

^"     a,' -a,'  ^  oT^^  ^     ^' 


2-i'=2i^  ^73^-  ''y  (21). 


*=  «.'-«.'  +^^57::^' (^^^■ 

/=5<.(^-2)=-26<.^"A^. (24). 

This  last  equation  gives  the  optical  eflfect  of  the  pressure  at  any  point.     The 
law  of  the  magnitude  of  this  quantity  is  the  inverse  square  of  the  radius,  as  in 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  47 

Case  I.  ;  but  the  direction  of  the  principal  axes  ia  different,  as  in  this  case  they 
are  parallel  and  perpendicular  to  the  radius.  The  dark  bands  seen  by  polarized 
Ught  wiU  therefore  be  parallel  and  perpendicular  to  the  plane  of  polarisation,  in- 
stead of  being  inclined  at  an  angle  of  45",  as  in  Case  I. 

By  substituting  in  Equations  (18)  and  (20),  the  values  of  p  and  q  given   in 
(22)  and  (23),  we  find  that  when  r  =  a,. 


hx      (l\(      ^aX-ct'h-X  .     2   /      a,%-a,%\   ] 


X       \9/x 


=  o(^  +  ~]  +  2{Ka,^-Ka,^) 


1/1         1 


.(25). 


,9/x      3m/  '     ^  '  '       '  'Ui,'-a,'\9fj,     3mJ 
r       9/x  \  a/  —  a/  /      3? 


When   r  =  a.,   -  ^  ^  fo4-2  ^4-^)  +  ^^^  (  -        ^._^.      '   '-o 


(26). 


~     VSft      3my  "^   '  a;  -  a,'  \  9/x  ^      3m     /      ^  cv  -  a,'  1,9/x  "^  3m/  J 

From  these  equations  it  appears  that  the  longitudinal  compression  of  cylin- 
dric  tubes  is  proportional  to  the  longitudinal  pressure  referred  to  unit  of  surface 
when  the  lateral  pressures  are  constant,  so  that  for  a  given  pressure  the  com- 
pression is  inversely  as  the  sectional  area  of  the  tube. 

These  equations  may  be  simplified  in  the  following  cases  : — 

1.  When  the  external  and  internal  pressures  are  equal,  or  h^  =  h^. 

2.  When  the  external  pressure  is  to  the  internal  pressure  as  the  square  of 
tlie  interior  diameter  is  to  that  of  the  exterior  diameter,   or  when  a^-h^  =  a^-h^. 

3.  When  the  cylinder  is  soHd,   or  when  a.  =  0. 

4.  When  the  solid  becomes  an  indefinitely  extended  plate  with  a  cylindric 
hole  in  it,   or  when  a^  becomes  infinite. 

5.  When  pressure  is  applied  only  at  the  plane  surfaces  of  the  solid  cylinder, 
and   the   cylindric   surface   is   prevented   from    expanding   by   being    inclosed   in   a 

strong  case,  or  when    —  =  0. 

6.  When  pressure  is  applied  to  the  cylindric  surface,  and  the  ends  are 
retained  at  an  invariable  distance,  or  when   —  =  0. 

X 


48 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 


1.     When  ^ji  =  A„  the  equations  of  compression  become 


\9fi'*"3mj"'"^  '\9ij.     3m 


(27). 


7  =  i('>+2^)  +  3i(^-<') 

When  hi  =  hi  =  o,  then 

Zx  _hr  _  \ 
X  ~  r  "  Sfi' 

The  compression  of  a  cylindrical  vessel  exposed  on  all  sides  to  the  same 
hydrostatic  pressure  is  therefore  independent  of  m,  and  it  may  be  shewn  that 
the  same  is  true  for  a  vessel  of  any  shape. 


2.     When  a,%  =  a^% 


^  \9yx  "^  3m/ 


Bx 

X 


7  =  |w  +  3l(3^--»)^ 


(28). 


In  this  case,  when  o  =  0,  the  compressions  are  independent  of  /x. 
3.     In  a  solid  cylinder,  aj  =  0, 

The  expressions  for  —  and  —  are  the  same  as  those  in  the  first  case,  when 
h^  —  hf 

When  the  lon^tudinal  pressure  o  vanishes, 

Bx 

X 


r       '  \9/x     3m/  ' 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 


49 


When  the  cylinder  ia  pressed  on  the  plane  sides  only, 

8x 


r         \9fi      dmj 


4.     When  the  solid  is  infinite,  or  when  a,  is  infinite, 


p  =  K--._a-(\-K) 


I=<o{p-q)=-^a.;{h,-h,) 
r       9/x  ^  '      3m  ^  ' 


(29). 


5.     When  8r  =  0  in  a  solid  cylinder, 

Zx  Zo 


6.    When 


X      2m  +  3/A 

So;  _       hr  _      2>h 
x~   *    r  ~  m  +  6iM 


.(30). 


Since  the  expression  for  the  efiect  of  a  longitudinal  strain  is 
Bx 


if  we  make 

VOL.  I. 


-=o(—  +  —) 
X         \9/i,      3m/ ' 


r,      9mu,       ^,         8x        1 

E  = ^  ,   then    —  =  o  ^^ 

m  +  6/x  cc         E 


(31). 


50  THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 

The  quantity  E  may  be  deduced  from  experiment  on  the  extension  of  wires 
or  rods  of  the  substance,  and  /x  is  given  in  terms  of  m  and  E  by  the  equation, 

„  =  _^!!L_ (32), 

^^^  ^  =  S (^^)' 

P  being  the  extending  force,  h  the  length  of  the  rod,  s  the  sectional  area, 
and  Bx  the  elongation,  which  may  be  determined  by  the  deflection  of  a  wire, 
as  in  the  apparatus  of  S'  Gravesande,  or  by  direct  measurement. 


Case  IV. 

The  only  known  direct  method  of  finding  the  compressibihty  of  liquids  is 
that  employed  by  Canton,  (Ersted,  Perkins,  Aime,  &c. 

The  liquid  is  confined  in  a  vessel  with  a  narrow  neck,  then  pressure  is 
applied,  and  the  descent  of  the  liquid  in  the  tube  is  observed,  so  that  the 
difference  between  the  change  of  volume  of  liquid  and  the  change  of  internal 
capacity  of  the  vessel  may  be  determined. 

Now,  since  the  substance  of  which  the  vessel  is  formed  is  compressible,  a 
change  of  the  internal  capacity  is  possible.  If  the  pressure  be  applied  only  to 
the  contained  liquid,  it  is  evident  that  the  vessel  will  be  distended,  and  the 
compressibihty  of  the  liquid  will  appear  too  great.  The  pressure,  therefore,  is 
commonly  applied  externally  and  internally  at  the  same  time,  by  means  of  a 
hydrostatic  pressure  produced  by  water  compressed  either  in  a  strong  vessel  or 
in  the  depths  of  the  sea. 

As  it  does  not  necessarily  follow,  from  the  equality  of  the  external  and 
internal  pressures,  that  the  capacity  does  not  change,  the  equilibrium  of  the 
vessel  must  be  determined  theoretically.  (Ersted,  therefore,  obtained  from  Poisson 
his  solution  of  the  problem,  and  applied  it  to  the  case  of  a  vessel  of  lead. 
To  find  the  cubical  elasticity  of  lead,  he  appUed  the  theory  of  Poisson  to  the 
numerical  results  of  Tredgold.  As  the  compressibility  of  lead  thus  found  was 
greater  than  that  of  water,  (Ersted  expected  that  the  apparent  compressibility 
of  water  in  a  lead  vessel  would  be  negative.  On  making  the  experiment  the 
apparent   compressibihty   was  greater   in  lead  than  in  glass.     The  quantity  found 


THE    EQUILIBRrcrM    OF    ELASTIC    SOLIDS.  51 

by   Tredgold   from    the  extension   of  rods  was  that  denoted  by  E,  and  the  value 
of  ft   deduced  from  E  alone   by   the  formulae  of   Poisson   cannot   be    true,  unless 

—  =  |-;   and   as   —    for  lead    is    probably   more   than   3,   the   calculated   compressi- 
bility is  much  too  great. 

A  similar  experiment  was  made  by  Professor  Forbes,  who  used  a  vessel  of 
caoutchouc.  As  in  this  case  the  apparent  compressibility  vanishes,  it  appears 
that  the  cubical  compressibihty  of  caoutchouc  is  equal  to  that  of  water. 

Some  who  reject  the  mathematical  theories  as  unsatisfactory,  have  conjec- 
tured that  if  the  sides  of  the  vessel  be  sufficiently  thin,  the  pressure  on  both 
sides  being  equal,  the  compressibility  of  the  vessel  will  not  affect  the  result. 
The  following  calculations  shew  that  the  apparent  compressibility  of  the  liquid 
depends  on  the  compressibility  of  the  vessel,  and  is  independent  of  the  thickness 
when  the  pressures  are  equal. 

A  hollow  sphere,  whose  external  and  internal  radii  are  a^  and  a,,  is  acted 
on  by  external  and  internal  normal  pressures  h^  and  K,  it  is  required  to  deter- 
mine the  equilibrium  of  the  elastic  solid. 

The  pressures  at  any  point  in  the  solid  are : — 

1.  A  pressure  p  in  the  direction  of  the  radius. 

2.  A  pressure  q  in  the  perpendicular  plane. 

These  pressures  depend  on  the  distance  from  the  centre,  which  is  denoted 
by  r. 

The  compressions  at  any  point  are  -.—  in  the  radial  direction,  and  —  in 
the  tangent  plane,  the  values  of  these  compressions  are  : — 

fr=[h-^^P^''i)*h^ ('")• 

T  =  fe-3fJ(^  +  2,)  +  l5 (35). 

Multiplying  the  last  equation  by  r,  differentiating  with  respect  to  r,  and 
equating  the  result  with  that  of  the  first  equation,  we  find 


52 


THE    EQUILIBRITTM    OF    ELASTIC    SOLIDS. 


Since  the  forces  whicli   act   on   the   particle   in  the   direction   of  the   radius 
must  balance  one  another,  or 


2qdrde  +p  (rdey  =(^p  +  ^d7^(r  +  dry  6, 


_r  dp 


therefore  ^""-^  =  2  37 ^^^^' 

Substituting  this  value  of  q  -p  in  the  preceding  equation,  and  reducing, 

therefore 


^  +  2^  =  0. 
dr        dr 


Integrating, 
But 
and  the  equation  becomes 

therefore 


p-\-2q  =  c,. 
r  dp  , 


dp 
dr 


+  3^-^-i  =  0, 


1      c. 


Since    p  =  h,    when    r  =  a.,,   and  p  =  K   when   r  =  a,,   the   value   of  p   at   any 
distance  is  found  to  be 


^~     a^-af         r'    a^-a,' 
9-    a,'-ai    "^^    7^    <-a/ 


(37). 
.(38). 


When  r  =  a„  -y  =  -^r:^^  -  +  t  ^^  ^^737^3  ^ 

~  a,'  -  a/  U       2»i/      a/  -  «/  \jx      2wi/  _ 
When  the  external  and  internal  pressures  are  equal 


.(39). 


h^  =  h.,=p  =  q,  and  -y- 


SV     K 


.(40), 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  53 

the   change   of  internal  capacity  depends  entirely  on  the  cubical  elasticity  of  the 
vessel,  and  not  on  its  thickness  or  linear  elasticity. 

When   the   external   and  internal  pressures  are  inversely  as  the   cubes  of  the 
radii  of  the  surfaces  on  which  they  act, 


aX  =  a,%,  p  =  ^  K  q=  -i^K 


when  r  =  r-  —      ^     ' 


(41). 


V  2   ^^ 

In   this  case  the  change  of  capacity  depends  on  the  linear  elasticity  alone. 

M.  Regnault,  in  his  researches  on  the  theory  of  the  steam  engine,  has 
given  an  account  of  the  experiments  which  he  made  in  order  to  determine 
with  accuracy  the  compressibility  of  mercury. 

He  considers  the  mathematical  formulae  very  uncertain,  because  the  theories 
of  molecular  forces  from  which  they  are  deduced  are  probably  far  from  the 
truth ;  and  even  were  the  equations  free  from  error,  there  would  be  much 
uncertainty  in  the  ordinary  method  by  measuring  the  elongation  of  a  rod  of 
the  substance,  for  it  is  diflScult  to  ensure  that  the  material  of  the  rod  is  the 
same  as  that  of  the  hollow  sphere. 

He  has,  .therefore,  availed  himself  of  the  results  of  M.  Lam6  for  a  hollow 
sphere  in  three  different  cases,  in  the  first  of  which  the  pressure  acts  on  the 
interior  and  exterior  surface  at  the  same  time,  while  in  the  other  two  cases 
the  pressure  is  applied  to  the  exterior  or  interior  surface  alone.  Equation  (39) 
becomes  in  these  cases, — 

1.  When  ^1  = /ij, -^  =  —  and  the  compressibility  of  the  enclosed  liquid  being 
/x,,  and  the  apparent  diminution  of  volume  S'F, 

v-.£-;) «■ 

2.  When  /i,  =  0, 


54  THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 

3.     When  h,^0, 

8V_      h       K  ,  9^\ 

V      a^-a^  \ii      ^  m     ^  '         V2  J 

M.  Lamp's  equations  differ  from  these  only  in  assuming  that  fi,  =  |-m.  If 
this  assumption  be  correct,  then  the  coefficients  /u,,  m,  and  jMj,  may  be  found 
from  two  of  these  equations ;  but  since  one  of  these  equations  may  be  derived 
from  the  other  two,  the  three  coefficients  cannot  be  found  when  /u,  is  supposed 
independent  of  m.  In  Equations  (39),  the  quantities  which  may  be  varied  at 
pleasure  are  \  and  h^,  and  the  quantities  which  may  be  deduced  from  the 
apparent  compressions  are, 


'■=G+4)^°<^S-i)=^" 


therefore  some  independent  equation  between  these  quantities  must  be  found, 
and  this  cannot  be  done  by  means  of  the  sphere  alone;  some  other  experiment 
must  be  made  on  the  liquid,  or  on  another  portion  of  the  substance  of  which 
the  vessel  is  made. 

The  value  of  /x^,  the  elasticity  of  the  liquid,   may  be  previously  known. 

The  linear  elasticity  m  of  the  vessel  may  be  found  by  twisting  a  rod  of 
the  material  of  which  it  is  made ; 

Or,    the   value   of  E  may   be  found   by   the   elongation   or   bending   of    the 


We  have  here  five  quantities,  which  may  be  determined  by  experiment. 

on  sphere. 


,  audi: 

-i^ 

2 
3m 

We  have  here 

fiv 

(43) 

1. 

(42) 

2. 

(31) 

3. 

(17) 

4. 
5. 

+  —  )   by   external  pressure 


Cj  =  ( j    equal   pressures. 

m  by  twisting  the  rod. 

/Xj  the  elasticity  of  the  liquid. 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 


55 


When    the    elastic    sphere     is    solid,    the    internal    radius    a,    vanishes,    and 

fh=p  =  q,  and  -y  =  ^- 

When   the   case  becomes  that   of  a   spherical   cavity  in  an  infinite  solid,  the 
external  radius  a^  becomes  infinite,  and 


P=K-f{K-K) 


r- 

=  K+i 

^i'h-h,) 

r 

=  ^^>i+^^(^>-^^) 

1 
m 

v  = 

■'-! 

(44). 


The  effect  of  pressure  on  the  surface  of  a  spherical  cavity  on  any  other  part 
of  an  elastic  solid  is  therefore  inversely  proportional  to  the  cube  of  its  distance 
from  the  centre  of  the  cavity. 

When  one  of  the  surfaces  of  an  elastic  hollow  sphere  has  its  radius  rendered 
invariable  by  the  support  of  an  incompressible  sphere,  whose  radius  is  Oj,   then 

—  =  0,    when  r  =  a^, 


therefore 


2771 


q=h 


2a^m  +  3«//x 

3a,V 
2a>i  +  3a//x 


r*    2a^m  +  3a//x 


IK 


W hen  r  =  a,,    j-y  —  lu  r-—. ~—  ,- 

"    V       -2a>2  +  3a.//i, 


K^ 


1 


r*    2a/m  +  3a//i 


(45). 


Case  V. 

On    the    equilibrium    of    an    elastic    beam    of    rectangular   section    uniformly 
bent. 

By   supposing   the   bent   beam   to   be  produced  till  it  returns  into  itself,  we 
may  treat  it  as  a  hollow  cylinder. 


66  THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 

Let  a  rectangular  elastic  beam,  whose  length  is  2irc,  be  bent  into  a  circular 
form,  so  as  to  be  a  section  of  a  hollow  cylinder,  those  parts  of  the  beam  which 
lie  towards  the  centre  of  the  circle  will  be  longitudinally  compressed,  while  the 
opposite  parts  will  be  extended. 

The  expression   for  the  tangential  compression  is  therefore 

Br  _  r  —  c 
r  ~     c    ' 

r 


Sr 
Comparing  this  value  of  —  with  that  of  Equation  (20), 


V=(^-4)<''+-p+«)+^''' 


dr 

,,.       ,  /I         2\     ., 

ion 


and  by  (21),  q=p  +  r 

By   substituting  for  q  its  value,  and  dividing  by  r  (q-  +  ^)  •  the  equat: 

becomes 

dp     2m  +  3/x  j9  _  9?n/i.  —  {m  —  3/x)  o         9m/x     c 
dr       m  +  6fx  r~        (m  +  6fi)  r  (m  +  6/x)  r'  * 

a  linear  differential  equation,  which  gives 

^        ^  m  —  3fir  2m  +  3/x 

Ci   may  be  found   by   assumiQg  that   when   r^a^,  p  =  \,   and   q  may   be   found 
from  p  by  equation   (21). 

As  the  expressions  thus  found  are  long  and  cumbrous,  it  is  better  to  use 
the  following  approximations  : — 

_/_9m^\  y (     ) 

l^\llcl^  \ (48). 

In  these  expressions  a  is  half  the  depth  of  the  beam,  and  y  is  the  distance 
of  any  part  of  the  beam  from  the  neutral  surface,  which  in  this  case  is  a  cylin- 
dric  surface,  whose  radius  is  c. 

These  expressions  suppose  c  to  be  large  compared  with  a,  since  most  sub- 
stances break   when  -  exceeds  a  certain  small  quantity. 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  57 

Let   b  be   the   breadth  of  the   beam,   then   the   force   with  which   the  beam 
resists  flexure  =  M 


M=lhyq  =  ^^^-^  =  Ef (49), 


which  is  the  ordinary  expression  for  the  stiffness  of  a  rectangular  beam. 

The'  stiffness  of  a  beam  of  any  section,  the  form  of  which  is  expressed  by 
an  equation  between  x  and  y,  the  axis  of  x  being  perpendicular  to  the  plane  of 
flexure,  or  the  osculating  plane  of  the  axis  of  the  beam  at  any  point,  is  ex- 
pressed by 

Mc  =  E{ifdx (50), 

M  being  the  moment   of  the   force   which   bends  the  beam,  and  c  the  radius  of 
the  circle  into  which  it  is  bent. 

Case  YI. 

At  the  meeting  of  the  British  Association  in  1839,  Mr  James  Nasmyth 
described  his  method  of  making  concave  specula  of  silvered  glass  by  bending. 

A  circular  piece  of  silvered  plate-glass  was  cemented  to  the  opening  of  an 
iron  vessel,  from  which  the  air  was  afterwards  exhausted.  The  mirror  then 
became  concave,  and  the  focal  distance  depended  on  the  pressure  of  the  air. 

Buffon  proposed  to  make  burning- mirrors  in  this  way,  and  to  produce  the 
partial  vacuum  by  the  combustion  of  the  air  in  the  vessel,  which  was  to  be 
effected  by  igniting  sulphur  in  the  interior  of  the  vessel  by  means  of  a  burn- 
ing-glass. Although  sulphur  evidently  would  not  answer  for  this  purpose,  phos- 
phorus might;  but  the  simplest  way  of  removing  the  air  is  by  means  of  the 
air-pump.  The  mirrors  which  were  actually  made  by  Buffon,  were  bent  by 
means  of  a  screw  acting  on  the  centre  of  the  glass. 

To  find  an  expression  for  the  curvature  produced  in  a  flat,  circular,  elastic 
plate,  by  the  difference  of  the  hydrostatic  pressures  which  act  on  each  side 
of  it,— 

Let  t  be  the  thickness  of  the  plate,  which  must  be  small  compared  with 
its  diameter. 

Let  the  form  of  the  middle  surface  of  the  plate,  after  the  curvature  is 
produced,  be  expressed  by  an  equation  between  r,  the  distance  of  any  point 
from  the  axis,  or  normal  to  the  centre  of  the  plate,  and  x  the  distance  of 
the  point  from  the  plane  in  which  the  middle  of  the  plate  originally  was,  and  let 

ds=-^{dxY  +  {dr)\ 

VOL    I.  8 


58  THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 

Let  A,  be  the  pressure  on  one  side  of  the  plate,  and  h^  that  on  the  other. 

Let  p  and  q  be  the  pressures  in  the  plane  of  the  plate  at  any  point,  p 
acting  in  the  direction  of  a  tangent  to  the  section  of  the  plate  by  a  plane 
passing  through  the  axis,  and  q  acting  in  the  direction  perpendicular  to  that 
plane. 

By  equating  the  forces  which  act  on  any  particle  in  a  direction  parallel  to 
the  axis,  we  find 

^    drdx  ,  ^   dpdx  ,  ^      d^x  ^     ,,       j^dr 

By  making  p  =  0  when  r  =  0  in  this  equation,  when  integrated, 

p-l^l^^--'^-) ("^- 

The  forces  perpendicular  to  the  axis  are 

[drV     .   dpdr  ,  ^     d^r     .^      i\dx       ^     . 

Substituting  for  p  its  value,  the  equation  gives 

_      (^1  -  h^      idr  dr     dx\      (h^  -  h^      /dr  ds^d^^ds  ^r\       ,     . 
^"  t        ''[d'sdi'^d^)'^      2t      "^^[didxd^      dxd^)""^     ^' 

The  equations  of  elasticity  become 

dSs     (\         1  \  /    ^       h,  +  h\^p 

Differentiating  -j-  =  -^  (""''')'  ^^^  ^  ^^  *^^^® 

dhr  dr     dr  dSs 

dr  ~       ds     ds  ds  ' 

By  a  comparison  of  these  values  of  -t—  , 


ds 

dr\ 
ds)  \9iJ, 


,      t^rwl         1\/     ,      ,K  +  h\,qdrp^     (I         l\fdp,dq\ 


w  dr      as 


THE    EQUILIBRIUM    OF    ELASTIC    SOUDS.  59 

To  obtain   an   expression  for  the  curvature   of  the  plate  at  the  vertex,  let  a 

be   the   radius   of  curvature,    then,   as   an   approximation   to   the   equation   of  the 

plate,  let 

r» 
x  —  —  . 
2a 

By  substituting  the  value   of  a:   in  the  values   of  p  and  q,  and  in  the  equa- 
tion of  elasticity,  the  approximate  value  of  a  is  found  to  be 


a  = 


18m/x,  \-\-h^    m-  3/x 

.    1     c  1        "T"  '  T  7~  ~T~z ; — TT" 


.(53). 


^i-A,  lOm  +  51/x       A,-^2  lOw  +  51/t ' 

Since  the  focal  distance   of  the   mirror,  or  -,   depends   on   the   difference   of 

pressures,   a  telescope  on   Mr  Nasmyth's   principle  would  act  as  an  aneroid  baro- 
meter, the  focal  distance  varying  inversely  as  the  pressure  of  the  atmosphere. 

Case  VIL 

To  find  the  conditions  of  torsion  of  a  cylinder  composed  of  a  great  number 
of  parallel  wires  bound  together  without  adhering  to  one  another. 

Let  X  be  the  length  of  the  cylinder,  a  its  radius,  r  the  radius  at  any  point, 
hS  the  angle  of  torsion,  M  the  force  producing  torsion,  hx  the  change  of  length, 
and  P  the  longitudinal  force.  Each  of  the  wires  becomes  a  helix  whose  radius 
is  r,  its  angular  rotation  Zd,  and  its  length  along  the  axis  x-Zx. 


Its  length  is  therefore  {rZey 


— IJ 


and  the  tension  is  =  jE;  1 1  -    /[  1  -  - ] V r^  (-]']  . 

This  force,  resolved  parallel  to  the  axis,  is 


60  THE    EQUIUBRTCM    OF    ELASTIC    SOUDS. 

and  since  —  and  r —  are  small,  we  may  assume 

XX 

-"-{-l-n?)'} <">■ 

The  force,  when  resolved  in  the  tangential  direction,  is  approximately 

"-■^m'i-m '"> 

By  eliminating  —  between  (54)  and  (55)  we  have 

X 


M:     ^^' 


^ip.E.^m (56). 

X  24  \  a?/ 


When  P  =  0,  M  depends  on  the  sixth  power  of  the  radius  and  the  cube 
of  the  angle  of  torsion,  when  the  cylinder  is  composed  of  separate  filaments. 

Since  the  force  of  torsion  for  a  homogeneous  cylinder  depends  on  the 
fourth  power  of  the  radius  and  the  first  power  of  the  angle  of  torsion,  the 
torsion  of  a  wire  having  a  fibrous  texture  will  depend  on  both  these  laws. 

The  parts  of  the  force  of  torsion  which  depend  on  these  two  laws  may  be 
found  by  experiment,  and  thus  the  difference  of  the  elasticities  in  the  direction 
of  the  axis  and  in  the  perpendicular  directions  may  be  determined. 

A  calculation  of  the  force  of  torsion,  on  this  supposition,  may  be  found  in 
Young's  Mathematical  Principles  of  Natural  Philosophy;  and  it  \s  introduced 
here  to  account  for  the  variations  from  the  law  of  Case  II.,  which  may  be 
observed  in  a  twisted  rod. 


Case  VIII. 

It  is  well  known  that  grindstones  and  fly-wheels  are  often  broken  by  the 
centrifugal  force  produced  by  their  rapid  rotation.  I  have  therefore  calculated 
the  strains  and  pressure  acting  on  an  elastic  cylinder  revolving  round  its  axis, 
and  acted  on  by  the  centrifugal  force  alone. 


THE    EQUILIBBIUM    OF    ELASTIC    SOLIDa. 


61 


The  equation  of  the  equilibrium   of  a  particle   [see  Equation  (21)],   becomes 


dp      Air'k    , 


where  q  and  p  are  the  tangential  and  radial  pressures,  k  is  the  weight  in 
pounds  of  a  cubic  inch  of  the  substance,  g  is  twice  the  height  in  inches  that 
a  body  falls  in  a  second,   t  is  the  time  of  revolution  of  the  cylinder  in  seconds. 

By   substituting  the  value  of  q  and  ^  in  Equations  (19),  (20),  and  neglect- 
ing 0, 


-(i-3^)(«|-?-g)-M^S-f-^.^) 


which  gives 


1  TT^k 


2gt^\ 


1      ,    Tj'k 


2+^K  +  ^« 


(-"?) 


TT'k 

2gf 


^=-V  +  2^»(-2  +  f)^  +  c. 


(57). 


If  the   radii   of  the   surfaces  of  the   hollow  cylinder  be  a,   and  cu„    and  the 
pressures   actmg  on  them   h^  and   h^,  then  the  values  of  c^  and  c,  are 


(58). 


-f^'-(«--.')S(^-S. 

When  o,  =  0,  as  in  the  case  of  a  solid  cylinder,  c,  =  0,  and 

«  =  *'+0  {2('^  +  «.')  +  |(3'^-«,')} (59). 

When  A,  =  0,  and  r^a^, 

^  =  ^U-2) (60). 

When  q  exceeds  the  tenacity  of  the  substance  in  pounds  per  square  inch, 
the  cylinder  will  give  way;  and  by  making  q  equal  to  the  number  of  pounds 
which  a  square  inch  of  the  substance  will  support,  the  velocity  may  be  found 
at  which  the  bursting  of  the  cylinder  will  take  place. 


g2  THE    EQUILIBRIUM    OP    ELASTIC    SOLIDS. 

Since  I=ho>(q-p)  =  '^  (^-2\br',   a  transparent  revolving  cylinder,  when 

polarized  light  is  transmitted  parallel  to  the  axis,  will  exhibit  rings  whose 
diameters  are  as  the  square  roots  of  an  arithmetical  progression,  and  brushes 
parallel  and  perpendicular  to  the  plane  of  polarization. 

Case  IX. 

A  hollow  cylinder  or  tube  is  surrounded  by  a  medium  of  a  constant 
temperature  while  a  liquid  of  a  different  temperature  is  made  to  flow  through 
it.  The  exterior  and  interior  surfaces  are  thus  kept  each  at  a  constant  tem- 
perature till  the  transference  of  heat  through  the  cylinder  becomes  uniform. 

Let   V   be   the   temperature    at    any   point,    then    when    this    quantity    has 


reached  its  limit, 


rdv  _ 


v  =  Ci\ogr  +  Ci (61). 

Let   the  temperatures  at  the   surfaces  be   0^  and   0^,   and  the   radii   of   the 
surfaces  a,  and  a^,  then 

^        0^-0^  loga,0^-logaA 

^'""logaj-loga/     '~    loga^-loga^ 
Let    the   coeflBcient  of   linear    dilatation   of   the    substance    be  c,,    then  the 
proportional  dilatation  at  any  point  will  be  expressed  by  c,v,  and  the  equations 
of  elasticity  (18),  (19),  (20),  become 

r      \,9/x     3m/  ^      ^    ^'    m 
The  equation  of  equHibrivuu  is 

2-P+r'^ (21), 

and  since  the  tube  is  supposed  to  be  of  a  considerable  length 

-J— =c^  a  constant  quantity. 

CL2C 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  63 

From  these  equations  we  find  ttat 

9/x      3m 
and  hence  v  =  c^\ogr  +  Cz,  p  may  be  found  in  terms  of  r. 

Hence  ?  =  (|l  +  4)  "  ^.«' •«§ '- ^.  ^  +  <'•  +  (|l  +  ^) ''.^- 

Since  I—hco  (q  —p)  =  ho)i—  +  - — )     CjCg  —  260)05  -^ , 

the   rings   seen   in  this   case   will  differ  from   those   described   in   Case   III.    only 
by  the  addition  of  a  constant  quantity. 

When  no   pressures   act  on  the   exterior   and   interior   surfaces   of   the    tube 
^j  =  ^„  =  0,  and 

/2    .  J_V^.^  Aoo-r  I  ^i'^/log^i-log«2  ,  a/logct,-a/logaA 


/^       1_\-       I  a^a^  log  g,  -  log  ct,     a^  log  a,  -  a/  log  a         \ 

^-1,9,.  + 3m/    ^^^3^^^S^       r^  a'-a^      +  <-a,^  +V' 

\9/x      3m/      '  '    \  r"         a{-a^      J 


...(62). 


There   will,   therefore,   be   no   action   on    polarized   light   for    the   ring   whose 
radius  is  r  when 

r"  =  2  „  log  -  . 


Case   X. 

Sir  David  Brewster  has  observed  {Edinburgh  Transacticms,  Vol.  viii.),  that 
when  a  solid  cylinder  of  glass  is  suddenly  heated  at  the  cylindric  siuface  a 
polarizing  force  is  developed,  which  is  at  any  point  proportional  to  the  square 
of    the   distance   from   the   axis   of    the    cylinder ;    that   is   to   say,    that   the   dif- 


64  THE    EQUILIBBIUM    OF    ELASTIC    SOLIDS. 

ference   of   retardation  of   the   oppositely  polari^ied  rays  of   %ht  is   proportional 
to  the  square  of  the  radius  r,  or 

/=  bCj^cor'  =  h(o  {q  —p)  =  hayr  -^  , 
Since  if  a  be  the  radius  of  the  cylinder,  ^  =  0  when  r^a, 

Hence  ?=J(3r'-o"). 

2 

By   substituting  these   values  of  p  and  q  in   equations   (19)   and   (20),   and 
,  .        d    h'       dhr    T  ^    , 


^=|(4  +  li)'-'  +  »"  (««)• 


c^  being  the   temperature   of  the  axis   of  the   cylinder,   and  c,  the   coefficient  of 
linear  expansion  for  glass. 

Case  XI. 

Heat  is  passing  uniformly  through  the  sides  of  a  spherical  vessel,  such  as 
the  ball  of  a  thermometer,  it  is  required  to  determine  the  mechanical  state  of 
the  sphere.  As  the  methods  are  nearly  the  same  as  in  Case  IX.,  it  will  be 
sufficient  to  give  the  results,  using  the  same  notation. 

,  dv  c, 

dr      ^'  *     r 

Ci  =  aM,— ?,    c-  =  -5-2 —, 

o,  —  o,  o,  —  a, 

1      /2    .1  \-^       1  . 

When  h,  =  h,  =  0  the  expression  for  p  becomes 

p  =  /2        ly-  r_aXLl      _^A.l^  a.'-a»         | 

^      \9/t*     3m/     '^  '      ''[a/-a/7^      a,-o^r  {0,-0,)  (o^-o^)]  ^     ' 

From  this  value  of  p  the  other  quantities  may  be  found,  as  in  Case  IX., 
from  the  equations  of  Case  IV. 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  65 


Case  XII. 

When  a  long  beam  is  bent  into  the  form  of  a  closed  circular  ring  (as  in 
Case  v.),  all  the  pressures  act  either  parallel  or  perpendicular  to  the  direction 
of  the  length  of  the  beam,  so  that  if  the  beam  were  divided  into  planks,  there 
would  be  no  tendency  of  the  planks  to  slide  on  one  another. 

But  when  the  beam  does  not  form  a  closed  circle,  the  planks  into  which  it 
may  be  supposed  to  be  divided  will  have  a  tendency  to  slide  on  one  another, 
and  the  amount  of  sliding  is  determined  by  the  linear  elasticity  of  the  sub- 
stance. The  deflection  of  the  beam  thus  arises  partly  from  the  bending  of  the 
whole  beam,  and  partly  from  the  sHding  of  the  planks ;  and  since  each  of  these 
deflections  is  small  compared  with  the  length  of  the  beam,  the  total  deflection 
will  be  the  sum  of  the  deflections  due  to  bending  and  sliding. 


Let 


A=Mc  =  E\xi/'dy (65). 


A  is  the  stiffiiess  of  the  beam  as  found  in  Case  Y.,  the  equation  of  the 
transverse  section  being  expressed  in  terms  of  x  and  y,  y  being  measured  from 
the  neutral  surface. 

Let  a  horizontal  beam,  whose  length  is  2l,  and  whose  weight  is  2w,  be 
supported  at  the   extremities  and  loaded  at  the  middle  with  a  weight   W. 

Let  the  deflection  at  any  point  be  expressed  by  h^,  and  let  this  quantity 
be   small   compared   with  the  length  of  the  beam. 

At  the  middle  of  the  beam,  8,y  is  found  by  the  usual  methods  to  be 

%  =  ^  {-h^w  +  ^^l'W)  (66). 


Let 


B  =  —  \xdy  =  —  (sectional  area) (jo7). 


B   is    the    resistance    of    the   beam   to   the   sliding   of  the   planks.      The   de- 
flection of  the  beam  arising  from  this  cause  is 

%  =  2]b(^'+^^ (68). 

VOL.  I.  9 


66  THE    EQXnUBRnJM    OF    ELASTIC    SOLIDS. 

This   quantity  is  small  compared   with  S^y,  when  the   depth   of  the   beam  is 
small  compared  with  its  length. 

The  whole  deflection  ^y  =  B^  +  S^ 


A3/  =  -  (^.Z-^iS  +  ^  {U  +^l) (^^)- 


Case  XIII. 

When  the  values  of  the  compressions  at  any  point  have  been  found,  when 
two  difierent  sets  of  forces  act  on  a  solid  separately,  the  compressions,  when 
the  forces  act  at  the  same  time,  may  be  found  by  the  composition  of  com- 
pressions, because  the  small  compressions  are  independent  of  one  another. 

It  appears  from  Case  I.,  that  if  a  cylinder  be  twisted  as  there  described, 
the  compressions  will  be  inversely  proportional  to  the  square  of  the  distance 
from  the  centre. 

If  two  cylindric  surfaces,  whose  axes  are  perpendicular  to  the  plane  of  an 
indefinite  elastic  plate,  be  equally  twisted  in  the  same  direction,  the  resultant 
compression  in  any  direction  may  be  found  by  adding  the  compression  due  to 
each  resolved  in  that  direction. 

The  result  of  this  operation  may  be  thus  stated  geometrically.  Let  A^  and 
A^  (Fig.  1)  be  the  centres  of  the  twisted  cylinders.  Join  ^1^25  and  bisect  A^A, 
in  0.     Draw  OBC  at  right  angles,  and  cut  off  OB^^  and  OB^  each  equal  to  OA^. 

Then  the  difference  of  the  retardation  of  oppositely  polarized  rays  of  light 
passing  perpendicularly  through  any  point  of  the  plane  varies  directly  as  the 
product  of  its  distances  from  B^  and  B^,  and  inversely  as  the  square  of  the 
product  of  its  distances  from  A^  and  A^. 

The  isochromatic  lines  are  represented  in  the  figure. 

The  retardation  is  infinite  at  the  points  ^1  and  A^;  it  vanishes  at  B^^ 
and  jBj  ;  and  if  the  retardation  at  0  be  taken  for  unity,  the  isochromatic  curves 
2,  4,  surround  Aj^  and  A^;  that  in  which  the  retardation  is  unity  has  two 
loops,  and  passes  through  0;  the  curves  ^,  ^  are  continuous,  and  have  points 
of    contrary    flexure ;    the    curve   ^   has    multiple    points    at    Cj   and    C,,    where 


THE    EQUILIBEIUM    OF    ELASTIC    SOLIDS. 


67 


.4,(7,  =  -4,^,,  and  two  loops  surrounding  B^  and  B^',  the  other  curves,  for  which 
/=l4-»  -gS-j  ^c-»  consist  each  of  two  ovals  surrounding  B^  and  jB,,  and  an 
exterior  portion  surrounding  all  the  former  curves. 

Fig.  1. 


I  have  produced  these  curves  in  the  jelly  of  isinglass  described  in  Case  I. 
They  are  best  seen  by  using  circularly  polarised  light,  as  the  curves  are  then 
seen  without  interruption,  and  their  resemblance  to  the  calculated  curves  is 
more  apparent.  To  avoid  crowding  the  curves  toward  the  centre  of  the  figure, 
I  have  taken  the  values  of  /  for  the  different  curves,  not  in  an  arithmetical, 
but  in  a  geometrical  progression,  ascending  by  powers  of  2. 


68 


THE    EQUILrBRTOM    OF    ELASTIC    SOLIDS. 


Case  XIV. 

On  the  determination  of  the  pressures  which  act  in  the  interior  of  trans- 
parent solids,  from  observations  of  the  action  of  the  solid  on  polarized  light. 

Sir  David  Brewster  has  pointed  out  the  method  by  which  polarized  light 
might  be  made  to  indicate  the  strains  in  elastic  solids ;  and  his  experiments  on 
bent  glass  confirm  the  theories  of  the  bending  of  beams. 

The  phenomena  of  heated  and  unannealed  glass  are  of  a  much  more  complex 
nature,  and  they  cannot  be  predicted  and  explained  without  a  knowledge  of  the 
laws  of  cooling  and  solidification,  combined  with  those  of  elastic  equilibrium. 

In  Case  X.  I  have  given  an  example  of  the  inverse  problem,  in  the  case 
of  a  cylinder  in  which  the  action  on  light  followed  a  simple  law ;  and  I  now 
go  on  to  describe  the  method  of  determuiing  the  pressures  in  a  general  case, 
applying  it  to  the  case  of  a  triangle  of  unannealed  plate-glass. 


D  D 


Fig.  3. 


The  lines  of  equal  intensity  of  the  action  on  Hght  are  seen  without 
interruption,  by  using  circularly  polarized  light.  They  are  represented  in  Fig.  2, 
where  A,  BBB,  DDD  are  the  neutral  points,  or  points  of  no  action  on  light, 
and   CCC,  EEE  are  the  points  where  that  action  is  greatest ;   and  the  intensity 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  69 

of  the  action  at  any  other  point  is  determined  by  its  position  with  respect  to 
the  isochromatic  curves. 

The  direction  of  the  principal  axes  of  pressure  at  any  point  is  found  by 
transmitting  plane  polarized  light,  and  analysing  it  in  the  plane  perpendicular 
to  that  of  polarization.  The  light  is  then  restored  in  every  part  of  the  triangle, 
except  in  those  points  at  which  one  of  the  principal  axes  is  parallel  to  the 
plane  of  polarization.  A  dark  band  formed  of  all  these  points  is  seen,  which 
shifts  its  position  as  the  triangle  is  turned  round  in  its  own  plane.  Fig.  3 
represents  these  curves  for  every  fifteenth  degree  of  inclination.  They  correspond 
to  the  lines  of  equal  variation  of  the  needle  in  a  magnetic  chart. 

From  these  curves  others  may  be  found  which  shall  indicate,  by  their  own 
direction,  the  direction  of  the  principal  axes  at  any  point.  These  curves  of 
direction  of  compression  and  dilatation  are  represented  in  Fig.  4 ;  the  curves 
whose  direction  corresponds  to  that  of  compression  are  concave  toward  the 
centre  of  the  triangle,  and  intersect  at  right  angles  the  curves  of  dilatation. 

Let   the   isochromatic  lines  in  Fig.  2  be  determined  by  the  equation 

<^,{x,y)  =  I-  =  (o{q-p)-, 

where  /  is  the  difference  of  retardation  of  the  oppositely  polarized  rays,  and 
q  and  p  the  pressures  in  the  principal  axes  at  any  point,  z  being  the  thick- 
ness of  the  plate. 

Let  the  lines  of  equal  inclination  be  determined  by  the  equation 

<^2  (^.  y)  =  tan  6, 

6  being  the  angle  of  inclination  of  the  principal  axes ;  then  the  differential 
equation  of  the  curves  of  direction  of  compression  and  dilatation  (Fig.  4)  is 

By  considering  any  particle  of  the  plate  as  a  portion  of  a  cylinder  whose 
axis  passes  through  the  centre  of  curvature  of  the  curve  of  compression,  we  find 

?-?>=^^ (21). 


70  THE    EQUILIBRIUM    OF    EliASTIC    SOLIDS. 

Let  R  denote  the  radius  of  curvature  of  the  curve  of  compression  at  any 
point,  and  let  S  denote  the  length  of  the  curve  of  dilatation  at  the  same 
point, 

and  since  {q  -p),  R  and  S  are  known,  and  since  at  the  surface,  where  (^^  {x,  y)  =  0, 
j9  =  0,  all  the  data  are  given  for  determining  the  absolute  value  of  p  by  inte- 
gration. 

Though  this  is  the  best  method  of  finding  p  and  q  by  graphic  construc- 
tion, it  is  much  better,  when  the  equations  of  the  curves  have  been  found,  that 
is,  when  ^i  and  <j>^  are  known,  to  resolve  the  pressures  in  the  direction  of  the 
axes. 

The  new  quantities  are  p^,  p„  and  ^3 ;  and  the  equations  are 

tan^=-^,    {p-qY  =  q.'  +  (p.-p.y,    Pi+P.=P  +  q- 
Pi    Pi 

It  is  therefore  possible  to  find  the  pressures  from  the  curves  of  equal  tint 
and  equal  inclination,  in  any  case  in  which  it  may  be  required.  In  the  mean- 
time the  curves  of  Figs.  2,  3,  4  shew  the  correctness  of  Sir  John  Herschell's 
ingenious  explanation  of  the  phenomena  of  heated  and  unannealed  glass. 


Note  A. 

As  the  mathematical  laws  of  compressions  and  pressures  have  been  very  thoroughly 
investigated,  and  as  they  are  demonstrated  with  great  elegance  in  the  very  complete  and 
elaborate  memoir  of  MM.  Lamd  and  Clapeyron,  I  shall  state  as  briefly  as  possible  their  results. 

Let  a  solid  be  subjected  to  compressions  or  pressures  of  any  kind,  then,  if  through  any 
point  in  the  solid  lines  be  drawn  whose  lengths,  measured  from  the  given  point,  are  pro- 
portional to  the  compression  or  pressure  at  the  point  resolved  in  the  directions  in  which  the 
lines  are  drawn,  the  extremities  of  such  lines  will  be  in  the  surface  of  an  ellipsoid,  whose 
centre  is  the  given  point. 

The  properties  of  the  system  of  compressions  or  pressures  may  be  deduced  from  those 
of  the  ellipsoid. 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS.  71 

There  are  three  diameters  having  perpendicular  ordinates,  which  are  called  the  principal 
axes  of  the  ellipsoid. 

Similarly,  there  are  always  three  directions  in  the  compressed  particle  in  which  there 
is  no  tangential  action,  or  tendency  of  the  parts  to  slide  on  one  another.  These  directions 
are  called  the  principal  axes  of  compression  or  of  pressure,  and  in  homogeneous  solids  they 
always  coincide  with  each  other. 

The  compression  or  pressure  in  any  other  direction  is  equal  to  the  sum  of  the  products 
of  the  compressions  or  pressures  in  the  principal  axes  multiplied  into  the  squares  of  the 
cosines  of  the  angles  which  they  respectively  make  with  that  direction. 


Note  B. 

The  fundamental  equations  of  this  paper  differ  from  those  of  Navier,  Poisson,  &c.,  only 
in  not  assuming  an  invariable  ratio  between  the  linear  and  the  cubical  elasticity;  but  since 
I  have  not  attempted  to  deduce  them  from  the  laws  of  molecular  action,  some  other  reasons 
must  be  given  for  adopting  them. 

The  experiments  from  which  the  laws  are  deduced  are — 

1st.  Elastic  solids  put  into  motion  vibrate  isochronously,  so  that  the  sound  does  not 
vary  with  the  amplitude  of  the  vibrations. 

2nd.  Regnault's  experiments  on  hollow  spheres  shew  that  both  linear  and  cubic  com- 
pressions are  proportional  to  the  pressures. 

3rd.  Experiments  on  the  elongation  of  rods  and  tubes  immersed  in  water,  prove  that 
the  elongation,  the  decrease  of  diameter,  and  the  increase  of  volume,  are  proportional  to  the 
tension. 

4th.  In  Coulomb's  balance  of  torsion,  the  angles  of  torsion  are  proportional  to  the 
twisting  forces. 

It  would  appear  from  these  experiments,  that  compressions  are  always  proportional  to 
pressures. 

Professor  Stokes  has  expressed  this  by  making  one  of  his  coefficients  depend  on  the 
cubical  elasticity,  Avhile  the  other  is  deduced  from  the  displacement  of  shifting  produced  by 
a  given  tangential  force. 

M.  Cauchy  makes  one  coefficient  depend  on  the  linear  compression  produced  by  a  force 
acting  in  one  direction,  and  the  other  on  the  change  of  volume  produced  by  the  same  force. 

Both  of  these  methods  lead  to  a  correct  result ;  but  the  coefficients  of  Stokes  seem  to 
have  more  of  a  real  signification  than  those  of  Cauchy ;  I  have  therefore  adopted  tiiose  of 
Stokes,  using  the  symbols  m  and  fi,  and  the  fundamental  equations  (4)  and  (5),  which  define 
them. 


72 


THE    EQUILIBRIUM    OF    ELASTIC    SOLIDS. 


Note  C. 

As  the  coefficient  <w,  which  determines  the  optical  effect  of  pressure  on  a  substance, 
varies  from  one  substance  to  another,  and  is  probably  a  function  of  the  linear  elasticity,  a 
determination  of  its  value  in  different  substances  might  lead  to  some  explanation  of  the 
action  of  media  on  light. 


This  paper  commenced  by  pointing  out  the  insufficiency  of  all  theories  of  elastic  solids, 
in  which  the  equations  do  not  contain  two  independent  constants  deduced  from  experiments. 
One  of  these  constants  is  common  to  liquids  and  solids,  and  is  called  the  modulus  of  cubical 
elasticity.  The  other  is  peculiar  to  solids,  and  is  here  called  the  modulus  of  linear  elasticity. 
The  equations  of  Navier,  Poisson,  and  Lam^  and  Clapeyron,  contain  only  one  coefficient; 
and  Professor  G.  G.  Stokes  of  Cambridge,  seems  to  have  formed  the  first  theory  of  elastic 
solids  which  recognised  the  independence  of  cubical  and  linear  elasticity,  although  M.  Cauchy 
seems  to  have  suggested  a  modification  of  the  old  theories,  which  made  the  ratio  of  linear 
to  cubical  elasticity  the  same  for  all  substances.  Professor  Stokes  has  deduced  the  theory 
of  elastic  solids  from  that  of  the  motion  of  fluids,  and  his  equations  are  identical  with  those 
of  this  paper,  which  are  deduced  from  the  two  following  assumptions. 

In  an  element  of  an  elastic  solid,  acted  on  by  three  pressures  at  right  angles  to  one 
another,  as  long  as  the  compressions  do  not  pass  the  limits  of  perfect  elasticity — 

1st.  The  sum  of  the  pressures,  in  three  rectangular  axes,  is  proportional  to  the  sum 
of  the  compressions  in  those  axes. 

2nd.  The  difference  of  the  pressures  in  two  axes  at  right  angles  to  one  another,  is 
proportional  to  the  difference  of  the  compressions  in  those  axes. 


Or,  in  symbols: 


(P.  +  P..i'J  =  3.(^%|4). 


(^.-^.)=-(l 


(P, 


p,)=r,j'y. 


(P,-P^  =  m 


fZz     Bx 


fi  being  the  modulus  of  auhical,  and  m  that  of  linear  elasticity. 

These   equations  are   found   to   be   very    convenient   for   the    solution   of   problems,   some 
of  which  were  given  in  the  latter  part  of  the  paper. 


THE    EQUILIBRIUM    OF     ELASTIC    SOLIDS.  73 

These  particular  cases  were — 

That  of  an  elastic  hollow  cylinder,  the  exterior  surface  of  which  was  fixed,  while  the 
interior  was  turned  through  a  small  angle.  The  action  of  a  transparent  solid  thus  twisted 
on  polarized  light,  was  calculated,  and  the  calculation  confirmed  by  experiment. 

The   second   case   related   to   the   torsion  of  cylindric   rods,   and   a  method   was   given   by 

which   m  may   be   found.     The   quantity   E=  ^  was  found   by  elongating,  or  by  bending 

the  rod  used  to  determine  m,  and  fi  is  found  by  the  equation, 

_      Em 
^~dm-6E' 

The  effect  of  pressure  on  the  surfaces  of  a  hollow  sphere  or  cylinder  was  calculated, 
and  the  result  applied  to  the  determination  of  the  cubical  compressibility  of  liquids  and 
solids. 

An  expression  was  found  for  the  curvature  of  an  elastic  plate  exposed  to  pressure  on 
one  side ;  and  the  state  of  cylinders  acted  on  by  centrifugal  force  and  by  heat  was 
determined. 

The  principle  of  the  superposition  of  compressions  and  pressures  was  applied  to  the  case  of 
a  bent  beam,  and  a  formula  was  given  to  determine  E  from  the  deflection  of  a  beam 
supported  at  both  ends  and  loaded  at  the  middle. 

The  paper  concluded  with  a  conjecture,  that  as  the  quantity  a  (which  expresses  the 
relation  of  the  inequality  of  pressure  in  a  solid  to  the  doubly-refracting  force  produced)  is 
probably  a  function  of  m,  the  determination  of  these  quantities  for  different  substances 
might  lead  to  a  more  complete  theory  of  double  refraction,  and  extend  our  knowledge  of  the 
laws  of  optics. 


VOL.  I.  10 


[Extracted   from   the  Cambridge   and  Dublin  Mathematical  Journal,  Vol.  viii.  p.  188, 

February/,  1854.] 


Solutions  of  Problems. 


1.  If  from  a  point  in  the  circumference  of  a  vertical  circle  two  heavy  particles  be  suc- 
cessively projected  along  the  curve,  their  initial  velocities  being  equal  and  either  in  the  same 
or  in  opposite  directions,  the  subsequent  motion  will  be  such  that  a  straight  line  joining 
the  particles  at  any  instant  will  touch  a  circle. 

Note.    The  particles  are  supposed  not  to  interfere  with  each  other's  motion. 


The  direct  analytical  proof  would  involve  the  properties  of  elliptic  integrals, 
but  it  may  be  made  to  depend  upon  the  following  geometrical  theorems. 

(1)  If  from  a  point  in  one  of  two  circles  a  right  line  be  drawn  cutting 
the  other,  the  rectangle  contained  by  the  segments  so  formed  is  double  of  the 
rectangle  contained  by  a  line  drawn  from  the  point  perpendicular  to  the  radical 
axis  of  the  two  circles,  and  the  line  joining  their  centres. 

The  radical  axis  is  the  line  joining  the  points  of  intersection  of  the  two 
circles.  It  is  always  a  real  hne,  whether  the  points  of  intersection  of  the  circles 
be  real  or  imaginary,  and  it  has  the  geometrical  property — that  if  from  any  point 
on  the  radical  axis,  straight  lines  be  drawn  cutting  the  circles,  the  rectangle  con- 
tained by  the  segments  formed  by  one  of  the  circles  is  equal  to  the  rectangle 
contained  by  the  segments  formed  by  the  other. 

The  analytical  proof  of  these  propositions  is  very  simple,  and  may  be  resorted 
to  if  a  geometrical  proof  does  not  suggest  itself  as  soon  as  the  requisite  figure 
is  constructed. 

If  ^,  B  be  the  centres  of  the  circles,  P  the  given  point  in  the  circle  whose 
centre   is   ^,    a   line  drawn   from   P   cuts   the   first   circle  in  p,  the  second  in  Q 


SOLUTIONS  OF  PROBLEMS.  75 

and  q,  and  the  radical  axis  in  R.     If  PH  be  drawn  perpendicular  to  the  radical 
axis,  then 

PQ.Pq  =  2AB.HP. 

CoR.  If  the  line  be  drawn  from  P  to  touch  the  circle  in  T,  instead  of 
cutting  it  in  Q  and  q,  then  the  square  of  the  tangent  PT  is  equal  to  the 
rectangle  2AB .  HP. 

Similarly,  if  ph  be  drawn  from  p  perpendicular  to  the  radical  axis 

p'P  =  2AB.hp. 

Hence,  if  a  line  be  drawn  touching  one  circle  in  T,  and  cutting  the  other 
in  P  and  p,  then 

(PTY  :  {pT)'  ::  HP  :  hp. 

(2)  If  two  straight  lines  touching  one  circle  and  cutting  another  be  made 
to  approach  each  other  indefinitely,  the  small  arcs  intercepted  by  their  inter- 
sections with  the  second  circle  wiU  be  ultimately  proportional  to  their  distances 
from  the  point  of  contact. 

This  result  may  easily  be  deduced  from  the  properties  of  the  similar 
triangles  FTP  and  ppT. 

Cor.  If  particles  P,  p  be  constrained  to  move  in  the  circle  A,  while 
the  line  Pp  joining  them  continually  touches  the  circle  B,  then  the  velocity 
of  P  at  any  instant  is  to  that  of  p  as  PT  to  pT ;  and  conversely,  if  the 
velocity  of  P  at  any  instant  be  to  that  of  P  as  PT  to  pT,  then  the  line 
Pp  will  continue  to  be  a  tangent  to  the  circle  B. 

Now  let  the  plane  of  the  circles  be  vertical  and  the  radical  axis  horizontal, 
and  let  gravity  act  on  the  particles  P,  p.  The  particles  were  projected  from 
the  same  point  with  the  same  velocity.  Let  this  velocity  be  that  due  to  the 
depth  of  the  point  of  projection  below  the  radical  axis.  Then  the  square  of 
the  velocity  at  any  other  point  will  be  proportional  to  the  perpendicular  from 
that  point  on  the  radical  axis  ;  or,  by  the  corollary  to  (l),  if  P  and  p  be  at 
any  time  at  the  extremities  of  the  line  PTp,  the  square  of  the  velocity  of  P 
will  be  to  the  square  of  the  velocity  of  p  as  PH  to  ph,  that  is,  as  (PTf  to 
(pTf.  Hence,  the  velocities  of  P  and  p  are  in  the  proportion  of  PT  to  pT, 
and  therefore,  by  the  corollary  to  (2),  the  line  joining  them  will  continue  a 
tangent  to  the  circle  B  during  each  instant,  and  will  therefore  remain  a  tangent 
during  the  motion. 


76  SOLUTIONS  OF  PROBLEMS. 

The  cb'cle  A,  the  radical  axis,  and  one  position  of  the  line  Pp,  are  given 
by  the  circumstances  of  projection  of  P  and  p.  From  these  data  it  is  easy  to 
determine  the  circle  jB  by  a  geometrical  construction. 

It  is  evident  that  the  character  of  the  motion  will  determine  the  position 
of  the  circle  B.  If  the  motion  is  oscillatory,  B  will  intersect  A.  If  P  and  p 
make  complete  revolutions  in  the  same  direction,  B  will  lie  entirely  within  A, 
but  if  they  move  in  opposite  directions,  B  will  lie  entirely  above  the  radical  axis. 

If  any  number  of  such  particles  be  projected  from  the  same  point  at  equal 
intervals  of  time  with  the  same  direction  and  velocity,  the  lines  joining  successive 
particles  at  any  instant  will  be  tangents  to  the  same  circle ;  and  if  the  time 
of  a  complete  revolution,  or  oscillation,  contain  n  of  these  intervals,  then  these 
lines  will  form  a  polygon  of  ?i  sides,  and  as  this  is  true  at  any  instant,  any 
number  of  such  polygons  may  be  formed. 

Hence,  the  following  geometrical  theorem  is  true : 

"If  two  circles  be  such  that  n  lines  can  be  drawn  touching  one  of  them 
and  having  their  successive  intersections,  including  that  of  the  last  and  first, 
on  the  circiunference  of  the  other,  the  construction  of  such  a  system  of  lines 
wiU  be  possible,  at  whatever  point  of  the  first  circle  we  draw  the  first  tangent." 


2.  A  transparent  medium  is  such  that  the  path  of  a  ray  of  light  within  it  is  a  given 
circle,  the  index  of  refraction  being  a  function  of  the  distance  from  a  given  point  in  the 
plane  of  the  circle. 

Find  the  form  of  this  function  and  shew  that  for  light  of  the  same  refrangibility — 

(1)  The  path  of  every  ray  witJdn  the  medium  is  a  circle, 

(2)  All  the  rays  proceeding  from  any  point  in  the  medium  will  meet  accurately  in 
another  point. 

(3)  If  rays  diverge  from  a  point  without  the  medium  and  enter  it  through  a  spherical 
surface  having  that  point  for  its  centre,  they  will  be  made  to  converge  accurately  to  a  point 
within  the  medium. 

Lemma  I.  Let  a  transparent  medium  be  so  constituted,  that  the  refractive 
index  is  the  same  at  the  same  distance  from  a  fixed  point,  then  the  path  of 
any    ray    of    light    within   the    medium    will  be   in   one   plane,    and  the  perpen- 


SOLUTIONS  OF  PROBLEMS.  77 

dicular    from    the    fixed   point   on   the    tangent   to   the   path   of  the   ray   at   any 
point  will  vary  inversely  as  the  refractive  index  of  the  medium  at  that  point. 

We  may  easily  prove  that  when  a  ray  of  light  passes  through  a  spherical 
surface,  separating  a  medium  whose  refractive  index  is  /x,  from  another  where 
it  is  /Aj,  the  plane  of  incidence  and  refraction  passes  through  the  centre  of 
the  sphere,  and  the  perpendiculars  on  the  direction  of  the  ray  before  and  after 
refraction  are  ir  the  ratio  of  /i,  to  fi^.  Since  this  is  true  of  any  number  of 
spherical  shells  of  different  refractive  powers,  it  is  also  true  when  the  index  of 
refraction  varies  continuously  from  one  shell  to  another,  and  therefore  the 
proposition  is  true. 

Lemma  II.  If  from  any  fixed  point  in  the  plane  of  a  circle,  a  perpen- 
dicular be  drawn  to  the  tangent  at  any  point  of  the  circumference,  the  rectangle 
contained  by  this  perpendicular  and  the  diameter  of  the  circle  is  equal  to  the 
square  of  the  line  joining  the  point  of  contact  with  the  fixed  point,  together 
with  the  rectangle  contained  by  the  segments  of  any  chord  through  the  fixed 
point. 

Let  APB  be  the  circle,  0  the  fixed  point;  then 
OY.FE=OP'  +  AO.OB, 


Produce  PO  to  Q,  and  join  QR,   then  the  triangles  OYP,  PQR  are  similar; 
therefore 

OY.PR=OP.PQ 

=  OP'  +  OP.OQ; 
.:   OY.PR=OP'  +  AO.OB. 
If  we  put  in  this  expression  AO .  OB  =  a^, 

PO  =  r,     OY=p,    PR  =  2p, 
it  becomes  2pp  =  ?'*+■  a*, 


78  SOLUTIONS  OF  PROBLEMS. 

To  find  the  law  of  the  index  of  refraction  of  the  medium,  so  that  a  ray 
from  A  may  describe  the  circle  APB,  /x  must  be  made  to  vary  inversely  as  p 
by  Lemma  I. 

Let   AO  =  r^,  and  let  the  refractive  index  at  A=fii;  then  generally 


h' 

_c 
p 

_  2C7p  . 
a'  +  r'' 

/^1  = 

.   2Cp 

a'  +  r:' 

a'  +  r,' 

but  at  A 

therefore 

The   value  of  /n   at   any   point   is  therefore  independent   of  p,  the   radius  of 

the   given  circle;   so   that  the  same  law  of  refractive  index  will  cause  any  other 

ray    to    describe    another    circle,   for  which   the   value   of    a'  is  the  same.      The 

a^  .       .  . 

value   of  OB   is   — ,   which   is   also   independent   of  p ;   so   that  every  ray  which 

proceeds  from  A  must  pass  through  B. 

Again,  if  we  assume  /x^  as  the  value  of  /x  when  r  =  0, 

ar  +  r,' 


therefore  h'  —  H-o 


d'  +  r'^-' 


a  result  independent  of  r^.  This  shews  that  any  point  A'  may  be  taken  as 
the  origin  of  the  ray  instead  of  A,  and  that  the  path  of  the  ray  will  still  be 
circular,  and  will  pass  through  another  point  B'  on  the  other  side  of  0,  such  that 

Next,  let  CP   be   a  ray   from   C,  a  point  without  the  medium,  falling  at  P 
on  a  spherical  surface  whose  centre  is  C. 

Let   0  be  the  fixed  point  in  the  medium  as  before.     Join  PO,  and  produce 

to    Q   till    OQ  =  jyp.     Through    Q   draw  a  circle  touching   CP  in  P,  and  cutting 

CO  in  A  and  B ;  then   PBQ  is  the  path  of  the  ray  within  the  medium. 


SOLUTIONS  OF  PROBLEMS.  79 


Since  CP  touches  the  circle,  we  have 
CP'^CA.  CB, 

=  {CO-OA){CO-\-OB); 

but  0A=  -^; 

therefore  CF'  =  CQ  +  CO  (oB  -  ^^ 

an    equation    whence     OB    may    be    found,    B   being   the   point   in   the   medium 
through  which  all  rays  from  C  pass. 

Note.  The  possibility  of  the  existence  of  a  medium  of  this  kind  possessing 
remarkable  optical  properties,  was  suggested  by  the  contemplation  of  the  structure 
of  the  crystalline  lens  in  fish;  and  the  method  of  searching  for  these  properties 
was  deduced  by  analogy  from  Newton's  Principia,  Lib.  L  Prop.  vii. 

It  would  require  a  more  accurate  investigation  into  the  law  of  the  refractive 
index  of  the  different  coats  of  the  lens  to  test  its  agreement  with  the  supposed 
medium,  which  is  an  optical  instrument  theoretically  perfect  for  homogeneous 
light,  and  might  be  made  achromatic  by  proper  adaptation  of  the  dispersive 
power  of  each  coat. 

On  the  other  hand,  we  find  that  the  law  of  the  index  of  refraction  which 
would  give  a  minimum  of  aberration  for  a  sphere  of  this  kind  placed  in  water, 
gives  results  not  discordant  with  facts,  so  far  as  they  can  be  readily  ascertained. 


[From  the  Transactions  of  the  Cambridge  Philosophical  Society,  Vol.  ix.  Part  iv.] 


IV.      On    the    Transformation    of  Surfaces    by    Bending. 


Euclid  has  given  two  definitions  of  a  surface,  which  may  be  taken  as 
examples  of  the  two  methods  of  investigating  their  properties. 

That  in  the  first  book  of  the  Elements  is — 

"A  superficies  is  that  which  has  only  length  and  breadth." 

The  superficies  difiers  from  a  line  in  having  breadth  as  well  as  length, 
and  the  conception  of  a  third  dimension  is  excluded  without  being  expHcitly 
introduced. 

In   the   eleventh  book,   where    the    definition    of    a    soHd    is    first    formally 
given,  the  definition  of  the  superficies  is  made  to  depend  on  that  of  the  solid — 
"  That  which  bounds  a  soHd  is  a  superficies." 

Here  the  conception  of  three  dimensions  in  space  is  employed  in  forming 
a  definition  more  perfect  than  that  belonging  to  plane  Geometry. 

In  our  analytical  treatises  on  geometry  a  surface  is  defined  by  a  function 
of  three  independent  variables  equated  to  zero.  The  surface  is  therefore  the 
boundary  between  the  portion  of  space  in  which  the  value  of  the  function  is 
positive,  and  that  in  which  it  is  negative;  so  that  we  may  now  define  a 
surface  to  be  the  boundary  of  any  assigned  portion   of  space. 

Surfaces  are  thus  considered  rather  with  reference  to  the  figures  which  they 
limit  than  as  having  any  properties  belonging  to  themselves. 

But  the  conception  of  a  surface  which  we  most  readily  form  is  that  of 
a   portion   of  matter,    extended   in   length   and   breadth,    but  of  which  the  thick- 


TKANSFORMATION  OF  SURFACES  BY  BENDING.  81 

ness   may    be    neglected.       By   excluding   the   thickness   altogether,    we   arrive   at 
Euclid's  first  definition,  which  we  may  state  thus — 

"  A  surface  is  a  lamina  of  which  the  thickness  is  diminished  so  as  to  become 
evanescent." 

We  are  thus  enabled  to  consider  a  surface  by  itself,  without  reference  to 
the  portion  of  space  of  which  it  is  a  boundary.  By  drawing  figures  on  the 
surface,  and  investigating  their  properties,  we  might  construct  a  system  of 
theorems,  which  would  be  true  independently  of  the  position  of  the  surface  in 
space,  and  which  might  remain  the  same  even  when  the  form  of  the  solid  of 
which  it  is  the  boundary  is  changed. 

When  the  properties  of  a  surface  with  respect  to  space  are  changed,  while 
the  relations  of  lines  and  figures  in  the  surface  itself  are  unaltered,  the  surface 
may  be  said  to  preserve  its  identity,  so  that  we  may  consider  it,  after  the 
change  has  taken  place,  as  the  same  surface. 

When  a  thin  material  lamina  is  made  to  assume  a  new  form  it  is  said 
to  be  hent.  In  certain  cases  this  process  of  bending  is  called  development,  and 
when  one  surface  is  bent  so  as  to  coincide  with  another  it  is  said  to  be 
applied  to  it. 

By  considering  the  lamina  as  deprived  of  rigidity,  elasticity,  and  other 
mechanical  properties,  and  neglecting  the  thickness,  we  arrive  at  a  mathemati- 
cal definition  of  this  kind  of  transformation. 

"  The  operation  of  bending  is  a  continuous  change  of  the  form  of  a  surface, 
without  extension  or  contraction  of  any  part  of  it." 

The  following  investigations  were  undertaken  with  the  hope  of  obtaining 
more  definite  conceptions  of  the  nature  of  such  transformations  by  the  aid  of 
those  geometrical  methods  which  appear  most  suitable  to  each  particular  case. 
The  order  of  arrangement  is  that  in  which  the  different  parts  of  the  subject 
presented  themselves  at  first  for  examination,  and  the  methods  employed  form 
parts  of  the  original  plan,  but  much  assistance  in  other  matters  has  been 
derived  from  the  works  of  Gauss*,  Liouvillef,  Bertrand^,  Puiseux§,  &c.,  references 
to  which  will  be  given  in  the  course  of  the  investigation. 

*  Disquisitiones   generalea   circa  superficies  curvas.     Presented    to   the    Royal   Society  of    Gottingen, 
8th  October,   1827.     Commentationes  Recentiores,  Tom.  vi. 

t  Liouville's  Journal,  xii.  X  ^^'^-   ^^'^'  §  ^^"^■ 

VOL,  I.  11 


82 


TRANSFORMATION    OF    SURFACES    BY    BENDING. 


On  the  Bending  of  Surfaces  generated  hy  the  motion  of  a  straight  line  in  space. 


If  a  straight  line  can  be  drawn  in  any  surface,  we  may  suppose  that 
part  of  the  surface  which  is  on  one  side  of  the  straight  line  to  be  fixed, 
while   the    other   part    is   turned   about    the    straight    line    as    an   axis. 

In  this  way  the  surface  may  be  bent  about  any  number  of  generating  lines 
as   axes  successively,  till  the  form  of  every  part  of  the  surface  is  altered. 

The  mathematical  conditions  of  this  kind  of  bending  may  be  obtained  in 
the  following  manner. 

Let  the  equations  of  the  generating  line  be  expressed  so  that  the  constants 
involved  in  them  are  functions  of  one  independent  variable  u,  by  the  variation  of 
which  we  pass  from  one  position  of  the  line  to  another. 

If  in  the  equations  of  the  generating  line  Aa,  u  =  u^,  then  in  the  equations 
of  the  line  Bh  we  may  put  u  =  U2,  and  from  the  equations  of  these  lines  we 
may  find  by  the  common  methods  the  equations  of  the  shortest  line  PQ  between 
Aa  and  Bb,  and  its  length,  which  we  may  call  S^.  We  may  also  find  the 
angle  between  the  directions  of  ^a  and  Bb,  and  let  this  angle  be  SO. 

In  the  same  way  from  the  equations  of 
Cc,  in  which  u  =  u^,  we  may  deduce  the  equa- 
tions of  RS,  the  shortest  line  between  Bb  and 
Cc,  its  length  8^5  and  the  angle  hd^  between 
the  directions  of  Bb  and  Cc.  We  may  also 
find  the  value  of  QR,  the  distance  between 
the  points  at  which  PQ  and  RS  cut  Bb. 
Let  QR  =  h(T,  and  let  the  angle  between  the 
directions  of  PQ  and  RS  be  S^. 

Now  suppose  the  part  of  tlie  surface  between  the  lines  Aa  and  Bb  to  be 
fixed,  while  the  part  between  Bb  and  Cc  is  turned  round  Bb  as  an  axis.  The 
line  RS  wiU  then  revolve  round  the  point  R,  remaining  perpendicular  to  Bhy 
and  Cc  will  still  be  at  the  same  distance  from  Bb,  and  wiU  make  the  same 
angle  with  it.  Hence  of  the  four  quantities  S4j  S^2>  ^cr  and  8</>,  8^  alone  will 
be  changed  by  the  process  of  bending.  8<^,  however,  may  be  varied  in  a 
perfectly  arbitrary  manner,  and  may   even  be   made   to   vanish. 


,•?_.. 


TRANSFORMATION     OF    SURFACES    BY    BENDING.  83 

For,  PQ  and  RS  being  both  perpendicular  to  Bh,  RS  may  be  turned 
about  Bh  till  it  is  parallel  to  PQ,  in  which  case  8^  becomes  =  0. 

By  repeating  this  process,  we  may  make  all  the  "  shortest  lines"  parallel  to 
one  another,  and  then  all  the  generating  lines  will  be  parallel  to  the  same 
plane. 

We  have  hitherto  considered  generating  lines  situated  at  finite  distances  from 
one  another ;  but  what  we  have  proved  will  be  equally  true  when  their  distances 
are  indefinitely  diminished.     Then  in  the  limit 


du 

B0 

dO 

u,-u, 

"         du 

Str 

da- 

"         du 

8(f> 

d(f> 

Uj  —  Wi         '*        du  ' 

All  these  quantities  being  functions  of  u,  ^,  0,  a-  and  (f),  are  functions  of  u 
and  of  each  other;  and  if  the  forms  of  these  functions  be  known,  the  positions 
of  all  the  generating  lines  may  be  successively  determined,  and  the  equation 
to  the  surface  may  be  found  by  integrating  the  equations  containing  the  values 
of  ^,   0,    a-  and  <j). 

When  the  surface  is  bent  in  any  manner  about  the  generating  lines,  C>  ^, 
and   a-  remain  unaltered,  but  cf)  is  changed  at  every  point. 

The  form  of  <^  as  a  function  of  u  will  depend  on  the  nature  of  the 
bending  ;  but  since  this  is  perfectly  arbitrary,  <^  may  be  any  arbitrary  function 
of  u.  In  this  way  we  may  find  the  form  of  any  surface  produced  by  bending 
the  given  surface  along  its  generating  lines. 

By  making  <f)  =  0,  we  make  all  the  generating  lines  parallel  to  the  same 
plane.  Let  this  plane  be  that  of  xy,  and  let  the  first  generating  line  coincide 
with  the  axis  of  x,  then  C  will  be  the  height  of  any  other  generating  line 
above  the  plane  of  xy,  and  0  the  angle  which  its  projection  on  that  plane 
makes  with  the  axis  of  x.  The  ultimate  intersections  of  the  projections  of  the 
generating  lines  on  the  plane  of  xy  will  form  a  curve,  whose  length,  measured 
from  the  axis  of  x,  will  be  o-. 


84  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

Since  ia  this  case  the  quantities  C>  ^,  and  cr  are  represented  bj  distinct 
geometrical  quantities,  we  may  simplify  the  consideration  of  all  surfaces  generated 
by  straight  lines  by  reducing  them  by  bending  to  the  case  in  which  those  lines 
are  parallel  to  a  given  plane. 

In   the   class   of  surfaces   in   which  the  generating  lines  ultimately  intersect, 

-T-  =  0,  and   ^  constant.     If  these   surfaces   be   bent   so   that   <j>  =  0,  the   whole   of 

the  generating  lines  will  lie  in  one  plane,  and  their  ultimate  intersections  will 
form  a  plane  curve.  The  surface  is  thus  reduced  to  one  plane,  and  therefore 
belongs  to  the  class  usually  described  as  "developable  surfaces."  The  form  of  a 
developable  surface  may  be  defined  by  means  of  the  three  quantities  0,  a-  and 
(f>.  The  generating  lines  form  by  their  ultimate  intersections  a  curve  of  double 
curvature  to  which  they  are  all  tangents.  This  curve  has  been  called  the 
cuspidal    edge.       The    length    of    this    curve   is   represented   by   a,    its    absolute 

curvature  at  any  point  by  -j-  ,  and  its  torsion  at  the  same  point  by  ■—  . 

When  the  surface  is  developed,  the  cuspidal  edge  becomes  a  plane  curve, 
and  every  part  of  the  surface  coincides  with  the  plane.  But  it  does  not  follow 
that  every  part  of  the  plane  is  capable  of  being  bent  into  the  original  form 
of  the  surface.  This  may  be  easily  seen  by  considering  the  surface  when  the 
position  of  the  cuspidal  edge  nearly  coincides  with  the  plane  curve  but  is  not 
confounded  with  it.  It  is  evident  that  if  from  any  point  in  space  a  tangent 
can  be  drawn  to  the  cuspidal  edge,  a  sheet  of  the  surface  passes  through  that 
point.  Hence  the  number  of  sheets  which  pass  through  one  point  is  the  same 
as  the  number  of  tangents  to  the  cuspidal  edge  which  pass  through  that 
point ;  and  since  the  same  is  true  in  the  limit,  the  number  of  sheets  which 
coincide  at  any  point  of  the  plane  is  the  same  as  the  number  of  tangents 
which  can  be  drawn  from  that  point  to  the  plane  curve.  In  constructing  a 
developable  surface  of  paper,  we  must  remove  those  parts  of  the  sheet  from 
which  no  real  tangents  can  be  drawn,  and  provide  additional  sheets  where  more 
than  one  tangent  can  be  drawn. 

In  the  case  of  developable  surfaces  we  see  the  importance  of  attending  to 
the  position  of  the  lines  of  bending;  for  though  all  developable  surfaces  may 
be  produced  from  the  same  plane  surface,  their  distinguishing  properties  depend 
on  the  form  of  the  plane   curve  which  determines  the  lines  of  bending. 


TRANSFORMATION     OF    SURFACES    BY    BENDING. 


85 


II. 

On  the  Bending  of  Surfaces  of  Revolution. 

In  the  cases  previously  considered,  the  bending  in  one  part  of  the  surface 
may  take  place  independently  of  that  in  any  other  part.  In  the  case  now 
before  us  the  bending  must  be  simultaneous  over  the  whole  surface,  and  its 
nature  must  be  investigated  by  a  different  method. 

The  position  of  any  point  P  on  a  surface  of  revolution  may  be  deter- 
mined by  the  distance  FV  from  the  vertex,  measured 
along  a  generating  line,  and  the  angle  AVO  which 
the  plane  of  the  generating  line  makes  with  a  fixed 
plane  through  the  axis.  Let  FV=s  and  AVO  =  6. 
Let  r  be  the  distance  {Pp)  of  P  from  the  axis ;  r 
will  be  a  function  of  s  depending  on  the  form  of  the 
generating  curve. 

Now  consider  the  small  rectangular  element  of  the  surface  at  P.  Its  length 
PR  =  Ss,  and  its  breadth  PQ  =  rhd,  where  r  is  a  function  of  s. 

If  in  another  surface  of  revolution  r  is  some  other  function  of  s,  then  the 
length  and  breadth  of  the  new  element  will  be  hs  and  rB$',  and  if 


r  =  /xr,     and    0'  =  -0, 

rze'=rze, 

and  the  dimensions  of  the  two  elements  will  be  the  same. 

Hence  the  one  element  may  be  applied  to  the  other,  and  the  one  surface 
may  be  applied  to  the  other  surface,  element  to  element,  by  bending  it.  To 
effect  this,  the  surface  must  be  divided  by  cutting  it  along  one  of  the  generating 
lines,  and  the  parts  opened  out,  or  made  to  overlap,  according  as  /x  is  greater 
or  less  than  unity. 

To  find  the  effect  of  this  transformation  on  the  form  of  the  surface  we 
must  find  the  equation  to  the  original  form  of  the  generating  line  in  terms  of 
6"  and  r,  then  putting  /  =  /ir,  the  equation  between  s  and  r  will  give  the  form 
of  the  generating  line  after  bending. 


86  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

When   /x  is  greater  than  1  it  may  happen   that  for   some   values  of  5,  y-  is 

greater  than  -.     In  this  case 

-j-  =  fi-j-  is  greater  than  1  ; 

a   result   which    indicates    that    the   curve   becomes    impossible   for  such  values  of 
s  and  ft. 

The  transformation  is  therefore  impossible  for  the  corresponding  part  of 
the  surface.  If,  however,  that  portion  of  the  original  surface  be  removed,  the 
remainder  may  be  subjected  to  the  required  transformation. 


The  theory  of  bending  when  apphed  to  the  case  of  surfaces  of  revolution 
presents  no  geometrical  difficulty,  and  little  variety;  but  when  we  pass  to 
the  consideration  of  surfaces  of  a  more  general  kind,  we  discover  the  insufficiency 
of  the  methods  hitherto  employed,  by  the  vagueness  of  our  ideas  with  respect 
to  the  nature  of  bending  in  such  cases.  In  the  former  case  the  bending  is 
of  one  kind  only,  and  depends  on  the  variation  of  one  variable ;  but  the 
surfaces  we  have  now  to  .consider  may  be  bent  in  an  infinite  variety  of  ways, 
depending  on  the  variation  of  three  variables,  of  which  we  do  not  yet  know  the 
nature  or  interdependence. 

We  have  therefore  to  discover  some  method  sufficiently  general  to  be  appli- 
cable to  every  possible  case,  and  yet  so  definite  as  to  limit  each  particular  case 
to  one  kind  of  bending  easily  imderstood. 

The  method  adopted  in  the  following  investigations  is  deduced  from  the 
consideration  of  the  surface  as  the  limit  of  the  inscribed  polyhedron,  when  the 
size  of  the  sides  is  indefinitely  diminished,  and  their  number  indefinitely  increased. 

A  method  is  then  described  by  which  such  a  polyhedron  may  be  inscribed 
in  any  surface  so  that  all  the  sides  shall  be  triangles,  and  aU  the  solid  angles 
composed  of  six  plane  angles. 

The  problem  of  the  bending  of  such  a  polyhedron  is  a  question  of  trigo- 
nometry, and  equations  might  be  found  connecting  the  angles  of  the  different 
edges  which  meet  in  each  soHd  angle  of  the  polyhedron.     It  will  be  shewn  that 


TRANSFORMATION    OF    SURFACES     BY     BENDING.  87 

the  conditions  thus  obtained  would  be  equivalent  to  three  equations  between 
the  six  angles  of  the  edges  belonging  to  each  solid  angle.  Hence  three  addi- 
tional conditions  would  be  necessary  to  determine  the  value  of  every  such  angle, 
and  the  problem  would  remain  as  indefinite  as  before.  But  if  by  any  means 
we  can  reduce  the  number  of  edges  meeting  in  a  point  to  four,  only  one  con- 
dition would  be  necessary  to  determine  them  all,  and  the  problem  would  be 
reduced  to  the  consideration  of  one  kind  of  bending  only. 

This  may  be  done  by  drawing  the  polyhedron  in  such  a  manner  that  the 
planes  of  adjacent  triangles  coincide  two  and  two,  and  form  quadrilateral  facets, 
four  of  which  meet  in  every  solid  angle.  The  bending  of  such  a  polyhedron 
can  take  place  only  in  one  way,  by  the  increase  of  the  angles  of  two  of  the 
edges  which  meet  in  a  point,  and  the  diminution  of  the  angles  of  the  other  two. 

The  condition  of  such  a,  polyhedron  being  inscribed  in  any  surface  is  then 
found,  and  it  is  shewn  that  when  two  forms  of  the  same  surface  are  given, 
a  perfectly  definite  rule  may  be  given  by  which  two  corresponding  polyhedrons 
of  this  kind  may  be  inscribed,  one  in  each  surface. 

Since  the  kind  of  bending  completely  defines  the  nature  of  the  quadrilateral 
polyhedron  which  must  be  described,  the  lines  formed  by  the  edges  of  the 
quadrilateral  may  be  taken  as  an  indication  of  the  kind  of  bending  performed 
on  the  surface. 

These  lines  are  therefore  defined  as  "  Lines  of  Bending." 

When  the  lines  of  bending  are  given,  the  forms  of  the  quadrilateral  facets 
are  completely  determined  ;  and  if  we  know  the  angle  which  any  two  adjacent 
facets  make  with  one  another,  we  may  determine  the  angles  of  the  three  edges 
which  meet  it  at  one  of  its  extremities.  From  each  of  these  we  may  find  the 
angles  of  three  other  edges,  and  so  on,  so  that  the  form  of  the  polyhedron 
after  bending  will  be  completely  determined  when  the  angle  of  one  edge  is  given. 
The  bending  is  thus  made  to  depend  on  the  change  of  one  variable  only. 

In  this  way  the  angle  of  any  edge  may  be  calculated  from  that  of  any 
given  edge ;  but  since  this  may  be  done  in  two  different  ways,  by  passing 
along  two  different  sets  of  edges,  we  must  have  the  condition  that  these  results 
may  be  consistent  with  each  other.  This  condition  is  satisfied  by  the  method 
of  inscribing  the  polyhedron.  Another  condition  will  be  necessary  that  tlie 
change  of  the  angle  of  any  edge  due  to  a  small  change  of  the  given  angle, 
produced  by  bending,  may  be  the  same  by  both  calculations.  This  is  the  con- 
dition   of   "  Instantaneous  Lines  of   Bending."     That    tliis    condition    mav    ccntinue 


88  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

to  be  satisfied  during  the  whole  process  we  must  have  another,  which  is  the 
condition  for  "  Permanent  Lines  of  Bending." 

The  use  of  these  lines  of  bending  in  simplifying  the  theory  of  surfaces  is 
the  only  part  of  the  present  method  which  is  new,  although  the  investigations 
connected  with  them  naturally  led  to  the  employment  of  other  methods  which 
had  been  used  by  those  who  have  already  treated  of  this  subject.  A  state- 
ment of  the  principal  methods  and  results  of  these  mathematicians  will  save 
repetition,  and  will  indicate  the  different  points  of  view  under  which  the 
subject  may  present  itself. 

The  first  and  most  complete  memoir  on  the  subject  is  that  of  M.  Gauss, 
already  referred  to. 

The  method  which  he  employs  consists  in  referring  every  point  of  the 
surface  to  a  corresponding  point  of  a  sphere  whose  radius  is  unity.  Normals 
are  drawn  at  the  several  points  of  the  surface  toward  the  same  side  of  it, 
then  lines  drawn  through  the  centre  of  the  sphere  in  the  direction  of  each  of 
these  normals  intersect  the  surface  of  the  sphere  in  points  corresponding  to 
those  points  of  the  original  surface  at  which  the  normals  were  drawn. 

If  any  line  be  drawn  on  the  surface,  each  of  its  points  will  have  a 
corresponding  point  on  the  sphere,  so  that  there  will  be  a  corresponding  Hne 
on  the  sphere. 

If  the  line  on  the  surface  return  into  itself,  so  as  to  enclose  a  finite  area 
of  the  surface,  the  corresponding  curve  on  the  sphere  will  enclose  an  area  on 
the  sphere,  the  extent  of  which  will  depend  on  the  form  of  the  surface. 

This  area  on  the  sphere  has  been  defined  by  M.  Gauss  as  the  measure  of 
the  "entire  curvature"  of  the  area  on  the  surface.  This  mathematical  quantity 
is  of  great  use  in  the  theory  of  surfaces,  for  it  is  the  only  quantity  connected 
with  curvature  which  is  capable  of  being  expressed  as  the  sum  of  all  its  parts. 

The  sum  of  the  entire  curvatures  of  any  number  of  areas  is  the  entire 
curvature  of  their  sum,  and  the  entire  curvature  of  any  area  depends  on  the 
form  of  its  boundary  only,  and  is  not  altered  by  any  change  in  the  form  of 
the  surface  within  the  boundary  line. 

The  curvature  of  the  surface  may  even  be  discontinuous,  so  that  we  may 
speak  of  the  entire  curvature  of  a  portion  of  a  polyhedron,  and  calculate  its 
amount. 

If  the  dimensions  of  the  closed  curve  be  diminished  so  that  it  may  be 
treated  as  an  element  of  the  surface,  the  ultimate  ratio  of  the  entire  curvature 


TRANSFORMATION     OF    SURFACES    BY    BENDING.  89 

to    the    area    of    the    element    on    the    surface  is   taken   as   the    measure    of  the 
"  specific  curvature  "  at  that  point  of  the  surface. 

The  terms  "entire"  and  "specific"  curvature  when  used  in  this  paper  are 
adopted  from  M.  Gauss,  although  the  use  of  the  sphere  and  the  areas  on  its 
surface  formed  an  essential  part  of  the  original  design.  The  use  of  these  terms 
will  save  much  explanation,  and  supersede  several  very  cumbrous  expressions. 

M.  Gauss  then  proceeds  to  find  several  analytical  expressions  for  the  measure 
of  specific  curvature  at  any  point  of  a  surface,  by  the  consideration  of  three 
points  very  near  each  other. 

The  co-ordinates  adopted  are  first  rectangular,  x  and  y,  or  x,  y  and  z,  being 
regarded  as  independent  variables. 

Then  the  points  on  the  surface  are  referred  to  two  systems  of  curves  drawn 
on  the  surface,  and  their  position  is  defined  by  the  values  of  two  independent 
variables  p  and  q,  such  that  by  varying  p  while  q  remains  constant,  we  obtain 
the  different  points  of  a  line  of  the  first  system,  while  p  constant  and  q 
variable  defines  a  line  of  the  second  system. 

By  means  of  these  variables,  points  on  the  surface  may  be  referred  to  lines 
on  the  surface  itself  instead  of  arbitrary  co-ordinates,  and  the  measure  of  cur- 
vature may  be  found  in  terms  of  p  and  q  when  the  surface  is  known. 

In  this  way  it  is  shewn  that  the  specific  curvature  at  any  point  is  the 
reciprocal  of  the  product  of  the  principal  radii  of  curvature  at  that  point,  a 
result  of  great  interest. 

From  the  condition  of  bending,  that  the  length  of  any  element  of  the 
curve  must  not  be  altered,  it  is  shewn  that  the  specific  curvature  at  any  point 
is  not  altered  by  bending. 

The   rest   of   the   memoir   is    occupied    with    the   consideration   of   particular 
modes  of  describing  the   two  systems   of  lines.     One   case   is   when   the   lines   of. 
the  first  system  are  geodesic,  or  "shortest"   lines  having  their  origin  in  a  point, 
and  the  second  system  is  drawn  so  as  to   cut   off  equal   lengths   from   the  curv^es 
of  the  first  system. 

The  angle  which  the  tangent  at  the  origin  of  a  line  of  the  first  system 
makes  with  a  fixed  line  is  taken  as  one  of  the  co-ordinates,  and  the  distance 
of  the  point  measured  along  that  line  as  the  other. 

It  is  shewn  that  the  two  systems  intersect  at  right  angles,  and  a  simple 
expression  is  found  for  the  specific  curvature  at  any  point. 

M.    Liouville  (Journal,   Tom.  xii.)    has   adopted  a   different   mode   of  simpli- 

VOL.  I.  22 


90  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

tying  the  problem.  He  has  shewn  that  on  every  surface  it  is  possible  to  find 
two  systems  of  curves  intersecting  at  right  angles,  such  that  the  length  and 
breadth  of  every  element  into  which  the  surface  is  thus  divided  shall  be  equal, 
and  that  an  infinite  number  of  such  systems  may  be  found.  By  means  of  these 
curves  he  has  found  a  much  simpler  expression  for  the  specific  curvature  than 
that  given  by  M.  Gauss. 

He  has  also  given,  in  a  note  to  his  edition  of  Monge,  a  method  of  testing 
two  given  surfaces  in  order  to  determine  whether  they  are  applicable  to  one 
another.  He  first  draws  on  both  surfaces  lines  of  equal  specific  curvature,  and 
determines  the  distance  between  two  corresponding  consecutive  lines  of  curvature 
in  both  surfaces. 

If  by  assuming  the  origin  properly  these  distances  can  be  made  equal  for 
every  part  of  the  surface,  the  two  surfaces  can  be  applied  to  each  other.  He 
has  developed  the  theorem  analytically,  of  which  this  is  only  the  geometrical 
interpretation. 

When  the  lines  of  equal  specific  curvature  are  equidistant  throughout  their 
whole  length,  as  in  the  case  of  surfaces  of  revolution,  the  surfaces  may  be 
applied  to  one  another  in  an  infinite  variety  of  ways. 

When  the  specific  curvature  at  every  point  of  the  surface  is  positive  and 
equal  to  a^,  the  surface  may  be  applied  to  a  sphere  of  radius  a,  and  when  the 
specific  curvature  is  negative  =  —a"  it  may  be  applied  to  the  surface  of  revo- 
lution which  cuts  at  right  angles  all  the  spheres  of  radius  a,  and  whose  centres 
are  in  a  straight  line. 

M.  Bertrand  has  given  in  the  Xlllth  Vol.  of  Liouville's  Journal  a  very 
simple  and  elegant  proof  of  the  theorem  of  M.  Gauss  about  the  product  of 
the  radii  of  curvature. 

He  supposes  one  extremity  of  an  inextensible  thread  to  be  fixed  at  a  point 
in  a  surface,  and  a  closed  curve  to  be  described  on  the  surface  by  the  other 
extremity,  the  thread  being  stretched  all  the  while.  It  is  evident  that  the 
length  of  such  a  curve  cannot  be  altered  by  bending  the  surface.  He  then 
calculates  the  length  of  this  curve,  considering  the  length  of  the  thread  small, 
and  finds  that  it  depends  on  the  product  of  the  principal  radii  of  curvature 
of  the  surface  at  the  fixed  point.  His  memoir  is  followed  by  a  note  of 
M.  Diguet,  who  deduces  the  same  result  from  a  consideration  of  the  area  of 
the  same  curve  ;  and  by  an  independent  memoir  of  M.  Puiseux,  who  seems  to 
give  the  same  proof  at  somewhat  greater  length. 


TRANSFORMATION    OF    SURFACES     BY     BENDING.  91 


Note.  Since  this  paper  was  written,  I  have  seen  the  Rev.  Professor  Jellett's  Memoir,  On 
the  Properties  of  Inextensible  Surfaces.  It  is  to  be  found  in  the  Transactions  of  the  Royal  Irish 
Academy,  Vol.  XXII.  Science,  &c.,  and  was  read  May  23,  18.53. 

Professor  Jellett  has  obtained  a  system  of  three  partial  differential  equations  which  express 
the  conditions  to  which  the  displacements  of  a  continuous  inextensible  membrane  are  subject. 
From  these  he  has  deduced  the  two  theorems  of  Gauss,  relating  to  the  invariability  of  the  product 
of  the  radii  of  curvature  at  any  point,  and  of  the  "  entire  curvature"  of  a  finite  portion  of  the 
surface. 

He  has  then  applied  his  method  to  the  consideration  of  cases  in  which  the  flexibihty  of  the 
surface  is  limited  by  certain  conditions,  and  he  has  obtained  the  following  results  : — 

If  the  displacements  of  an  inextensible  surface  he  all  parallel  to  the  same  plane,  the  mrface 
moves  as  a  rigid  body. 

Or,  more  generally, 

If  the  movement  of  an  inextensible  surface,  parallel  to  any  one  line,  be  that  of  a  rigid  body,  the 
entire  movement  is  that  of  a  rigid  body. 

The  following  theorems  relate  to  the  case  in  which  a  curve  traced  on  the  surface  is  rendered 
rigid  :— 

//  any  curve  be  traced  upon  an  inextensible  surface  whose  principal  radii  of  curvature  are  finite 
and  of  the  same  sign,  and  if  this  curve  he  rendered  immoveable,  the  entire  surface  will  become 
immoveable  also. 

In  a  developable  surface  composed  of  an  inextensible  membrane,  any  one  of  its  rectilinear 
sections  may  be  fixed  without  destroying  the  fiexibility  of  the  membrane. 

In  convexo-concave  surfaces,  there  are  two  directions  passing  through  every  point  of  the 
surface,  such  that  the  curvature  of  a  normal  section  taken  in  these  directions  vanishes.  We 
may  therefore  conceive  the  entire  surface  to  be  crossed  by  two  series  of  curves,  such  that 
a  tangent  drawn  to  either  of  them  at  any  point  shall  coincide  with  one  of  these  direc- 
tions. These  curves  Professor  Jellett  has  denominated  Curves  of  Flexure,  from  the  following 
properties : — 

Any  curve  of  fiexure  may  he  fi^ed  without  destroying  the  fiexibility  of  the  surface. 

If  an  arc  of  a  curve  traced  upon  an  inextensible  surface  be  rendered  fixed  or  rigid,  the  entire  of 
the  quadrilateral,  formed  by  drauring  the  two  curves  of  fiexure  through  each  extremity  of  the  curve, 
become  fixed  or  rigid  also. 

Professor  Jellett  has  also  investigated  the  properties  of  partially  inextensible  surfaces,  and 
of  thin  material  laminae  whose  extensibility  is  small,  and  in  a  note  he  has  demonstrated  the 
following  theorem : — 

If  a  closed  oval  surface  he  perfectly  inextensible,  it  is  also  perfectly  rigid. 

A  demonstration  of  one  of  Professor  Jellett's  theorems  will  be  found  at  the  end  of  this  paper. 

J.  C.  M. 
Aug.  30,   1851 


92  TRANSFORMATION    OF    SURFACES    BY    BENDING. 


On  the  properties  of  a  Surface  considered  as  the  limit  of  the  inscribed 

Polyhedron. 

1.  To  inscribe  a  polyhedron  in  a  given  surface,  aU  whose  sides  shall  he 
triangles,  and  all  whose  solid  angles  shall  he  hexahedral. 

On  the  given  surface  describe  a  series  of  curves 
according  to  any  assumed  law.  Describe  a.  second  series 
intersecting  these  in  any  manner,  so  as  to  divide  the 
whole  surface  into  quadrilaterals.  Lastly,  describe  a 
third  series  (the  dotted  lines  in  the  figure),  so  as  to 
pass  through  all  the  intersections  of  the  first  and  second 
series,  forming  the  diagonals  of  the  quadrilaterals. 

The  surface  is  now  covered  with  a  network  of  curvilinear  triangles.  The 
plane  triangles  which  have  the  same  angular  points  will  form  a  polyhedron 
fulfilling  the  required  conditions.  By  increasing  the  number  of  the  curves  in 
each  series,  and  diminishing  their  distance,  we  may  make  the  polyhedron 
approximate  to  the  surface  without  limit.  At  the  same  time  the  polygons 
formed  by  the  edges  of  the  polyhedron  will  approximate  to  the  three  systems 
of  intersecting  curves. 

2.  To  find  the  measure  of  the  ''entire  curvature"  of  a  solid  angle  of  the 
'polyhedron,  and  of  a  finite  portion  of  its  surface. 

From  the  centre  of  a  sphere  whose  radius  is  unity  draw  perpendiculars  to 
the  planes  of  the  six  sides  forming  the  solid  angle.  These  lines  will  meet  the 
surface  in  six  points  on  the  same  side  of  the  centre,  which  being  joined  by 
arcs  of  great  circles  will  form  a  hexagon  on  the  surface  of  the  sphere. 

The  area  of  this  hexagon  represents  the  entire  curvature  of  the  solid  angle. 

It  is  plain  by  spherical  geometry  that  the  angles  of  this  hexagon  are  the 
supplements  of  the  six  plane  angles  which  form  the  solid  angle,  and  that  the 
arcs  forming  the  sides  are  the  supplements  of  those  subtended  by  the  angles 
of  the  six  edges  formed  by  adjacent  sides. 

The  area  of  the  hexagon  is  equal  to  the  excess  of  the  sum  of  its  angles 
above  eight  right  angles,  or  to  the  defect  of  the  sum  of  the  six  plane  angles 
from    four    right    angles,     which    is    the    same    thing.      Since   these   angles  are 


TRANSFORMATION    OF    SURFACES    BY    BENDING.  93 

invariable,  the  bending  of  the  polyhedron  cannot  alter  the  measure  of  curvature 
of  each  of  its  solid  angles. 

If  perpendiculars  be  drawn  to  the  sides  of  the  polyhedron  which  contain 
other  solid  angles,  additional  points  on  the  sphere  will  be  found,  and  if  these 
be  joined  by  arcs  of  great  circles,  a  network  of  hexagons  will  be  formed  on 
the  sphere,  each  of  which  corresponds  to  a  solid  angle  of  the  polyhedron  and 
represents  its  "  entire  curvature." 

The  entire  curvature  of  any  assigned  portion  of  the  polyhedron  is  the  sum 
of  the  entire  curvatures  of  the  solid  angles  it  contains.  It  is  therefore  repre- 
sented by  a  polygon  on  the  sphere,  which  is  composed  of  all  the  hexagons 
corresponding  to  its  solid  angles. 

If  a  polygon  composed  of  the  edges  of  the  polyhedron  be  taken  as  the 
boundary  of  the  assigned  portion,  the  sum  of  its  exterior  angles  will  be  the 
same  as  the  sum  of  the  exterior  angles  of  the  polygon  on  the  sphere ;  but 
the  area  of  a  spherical  polygon  is  equal  to  the  defect  of  the  sum  of  its 
exterior  angles  from  four  right  angles,  and  this  is  the  measure  of  entire  curva- 
ture. 

Therefore  the  entire  curvature  of  the  portion  of  the  polyhedron  enclosed 
by  the  polygon  is  equal  to  the  defect  of  the  sum  of  its  exterior  angles  from 
four  right  angles. 

Since  the  entire  curvature  of  each  solid  angle  is  unaltered  by  bending, 
that   of  a  finite  portion  of  the  surface  must  be  also  invariable. 

3.  On  the  "  Conic  of  Contact,"  and  its  use  in  determining  the  curvature 
of  normal  sections  of  a  surface. 

Suppose  the  plane  of  one  of  the  triangular  facets  of  the  polyhedron  to 
be  produced  till  it  cuts  the  surface.  The  form  of  the  curve  of  intersection 
\7ill  depend  on  the  nature  of  the  surface,  and  when  the  size  of  the  triangle 
is  indefinitely  diminished,  it  will  approximate,  to  the  form  of  a  conic  section. 

For  we  may  suppose  a  surface  of  the  second  order  constructed  so  as  to 
have  a  contact  of  the  second  order  with  the  given  surface  at  a  point  within 
the  angular  points  of  the  triangle.  The  curve  of  intersection  with  this  surface 
will  be  the  conic  section  to  which  the  other  curve  of  intersection  approaches. 
This  curve  will  be  henceforth  called  the  "  Conic  of  Contact,"  for  want  of  a  better 
name. 


1)4 


TRANSFORMATION    OF    SURFACES    BY    BENDING. 


To  Jind  tJie  radius  of  curvature  of  a  normal  section 
of  the  surface. 

Let  ARa  be  the  conic  of  contact,  C  its  centre,  and 
CP  perpendicular  to  its  plane.  rPR  a  normal  section,  and 
0  its  centre  of  curvature,   then 

=  1.^   in   the  limit,  when  CR  and  PR  coincide, 
^  CP 

-s  CP' 
or  calling  CP  the  "sa,gitta,"  we  have  this  theorem: 

"The  radius  of  curvature  of  a  normal  section  is  equal  to  the  square  of 
the  corresponding  diameter  of  the  conic  of  contact  divided  by  eight  times  the 
sagitta." 


4.  To  insciihe  a  polyhedron  in  a  given  surface,  all  ivhose  sides  shcdl  he 
plane  quadrilaterals,  and  all  whose  solid  angles  shall   he  tetraliedral. 

Suppose  the  three  systems  of  curves  drawn  as  described  in  sect.  (1),  then 
each  of  the  quadrilaterals  formed  by  the  intersection  of  the  first  and  second 
systems  is  divided  into  two  triangles  by  the  third  system.  If  the  planes  of 
these  two  triangles  coincide,  they  form  a  plane  quadrilateral,  and  if  every  such 
pair  of  triangles  coincide,  the  polyhedron  will  satisfy  the  required  condition. 

Let  ahc  be  one  of  these  triangles,  and  acd  the 
other,  which  is  to  be  in  the  same  plane  with  ahc. 
Then  if  the  plane  of  ahc  be  produced  to  meet  the 
surface  in  the  conic  of  contact,  the  curve  will  pass 
through  ahc  and  d.  Hence  ahcd  must  be  a  quad- 
rilateral inscribed  in  the  conic  of  contact. 

But  since  ah  and  dc  belong  to  the  same  system  of  curves,  they  will  be 
ultimately  parallel  when  the  size  of  the  facets  is  diminished,  and  for  a  similar 
reason,  ad  and  ho  will  be  ultimately  parallel.  Hence  ahcd  will  become  a  paral- 
lelogram, but  the  sides  of  a  parallelogram  inscribed  in  a  conic  are  parallel  to 
conjugate  diameters. 


TRANSFORMATION  OF  SURFACES  BY  BENDING.  ©5 

Therefore  the  directions  of  two  curves  of  the  first  and  second  system  at 
their  point  of  intersection  must  be  parallel  to  two  conjugate  diameters  of  the 
conic  of  contact  at  that  point  in  order  that  such  a  polyhedron  may  be  inscribed. 

Systems  of  curves  intersecting  in  this  manner  will  be  referred  to  as  "conju- 
gate systems." 

5.  On  the  elementary  conditions  of  the  applicahilitij  of  two  surfaces. 

It  is  evident,  that  if  one  surface  is  capable  of  being  appUed  to  another  by 
bending,  every  point,  line,  or  angle  in  the  first  has  its  corresponding  point,  line, 
or  angle  in  the  second. 

If  the  transformation  of  the  surface  be  eflfected  without  the  extension  or 
contraction  of  any  part,  no  line  drawn  on  the  surface  can  experience  any  change 
in  its  length,  and  if  this  condition  be  fulfilled,  there  can  be  no  extension  or 
contraction. 

Therefore  the  condition  of  bending  is,  that  if  any  line  whatever  be  drawn 
on  the  first  surface,  the  corresponding  curve  on  the  second  surface  is  equal  to  it 
in  length.     All  other  conditions  of  bending  may  be  deduced  from  this. 

6.  If  two  curves  on  the  first  surface  intersect,  the  corresponcling  curves  on  the 
second  surface  intersect  at  the  same  angle. 

On  the  first  surface  draw  any  curve,  so  as  to  form  a  triangle  with  the 
curves  already  drawn,  and  let  the  sides  of  this  triangle  be  indefinitely  dimin- 
ished, by  making  the  new  curve  approach  to  the  intersection  of  the  former 
curves.  Let  the  same  thing  be  done  on  the  second  surface.  We  shall  then 
have  two  corresponding  triangles  whose  sides  are  equal  each  to  each,  by  (5), 
and  since  their  sides  are  indefinitely  small,  we  may  regard  them  as  straight 
lines.  Therefore  by  Euclid  i.  8,  the  angle  of  the  first  triangle  formed  by  the 
intersection  of  the  two  curves  is  equal  to  the  corresponding  angle  of  the  second. 

7.  At  any  given  point  of  the  first  surface,  two  directions  can  he  found,  which 
are  conjugate  to  each  other  with  respect  to  the  conic  of  contact  at  that  point,  and 
continue  to  he  conjugate  to  each  other  when  tJie  first  surface  is  transformed  into  the 
second. 

For  let  the  first  surface  be  transferred,  without  changing  its  form,  to  a 
position  such  that  the  given  point  coincides  with  the  corresponding  point  of  the 
second  surface,  and  the   normal   to   the   first   surface   coincides   with   that    of   the 


96 


TRANSFORMATION    OF    SURFACES    BY    BENDING. 


second  at  the  same  point.  Then  let  the  first  surface  be  turned  about  the  normal 
as  an  axis  till  the  tangent  of  any  line  through  the  point  coincides  with  the 
tangent  of  the  corresponding  line  in  the  second  surface. 

Then  by  (6)  any  pair  of  corresponding  lines  passing  through  the  point  will 
have  a  common  tangent,  and  will  therefore  coincide  in  direction  at  that  point. 

If  we  now  draw  the  conies  of  contact  belonging  to  each  surface  we  shall 
have  two  conies  with  the  same  centre,  and  the  problem  is  to  determine  a  pair 
of  conjugate  diameters  of  the  first  which  coincide  with  a  pair  of  conjugate 
diameters  of  the  second.  The  analytical  solution  gives  two  directions,  real, 
coincident,  or  impossible,  for  the  diameters  required. 

In  our  investigations  we  can  be  concerned  only  with  the  case  in  which  these 
directions  are  real. 

When  the  conies  intersect  in  four  points,  P,  Q,  R,  S,  FQES  is  a  parallelo- 
gram   inscribed   in    both    conies,    and  the   axes    CA,    CB, 
parallel  to  the  sides,  are  conjugate  in  both  conies. 

If  the  conies  do  not  intersect,  describe,  through  any 
point  P  of  the  second  conic,  a  conic  similar  to  and  con- 
centric with  the  first.  If  the  conies  intersect  in  four 
points,  we  must  proceed  as  before;  if  they  touch  in  two 
points,  the  diameter  through  those  points  and  its  conju- 
gate must  be  taken.  If  they  intersect  in  two  points  only, 
then  the  problem  is  impossible ;  and  if  they  coincide 
altogether,  the  conies  are  similar  and  similarly  situated, 
and  the  problem  is  indeterminate. 


8.  Two  surfaces  being  given  as  before,  one  pair  of  conjugate  systems  of 
curves  may  be  drawn  on  the  first  surface,  which  shall  correspond  to  a  pair  of 
conjugate  systems  on  the  second  surface. 

By  article  (7)  we  may  find  at  every  point  of  the  first  surface  two 
directions  conjugate  to  one  another,  corresponding  to  two  conjugate  directions  on 
the  second  surface.  These  directions  indicate  the  directions  of  the  two  systems 
of  curves  which  pass  through  that  point. 

Knowing  the  direction  which  every  curve  of  each  system  must  have  at  every 
point  of  its  course,  the  systems  of  curves  may  be  either  drawn  by  some  direct 
geometrical  method,  or  constructed  from  their  equations,  which  may  be  found  by 
solving  their  difierential  equations. 


TRANSFORMATION    OF    SURFACES    BY    BENDING.  97 

Two  systems  of  curves  being  drawn  on  the  first  surface,  the  corresponding 
systems  may  be  drawn  on  the  second  surface.  These  systems  being  conjugate 
to  each  other,  fulfil  the  condition  of  Art.  (4),  and  may  therefore  be  made  the 
means  of  constructing  a  polyhedron  with  quadrilateral  facets,  by  the  bending  of 
which  the  transformation  may  be  effected. 

These  systems  of  curves  will  be  referred  to  as  the  "first  and  second  systems 
of  Lines  of  Bending." 

9.     General  considerations  applicable  to  Lines  of  Bending. 

It  has  been  shewn  that  when  two  forms  of  a  surface  are  given,  one  of 
which  may  be  transformed  into  the  other  by  bending,  the  nature  of  the  Hnes 
of  bending  is  completely  determined.  Supposing  the  problem  reduced  to  its 
analyticid  expression,  the  equations  of  these  curves  would  appear  under  the 
form  of  double  solutions  of  differential  equations  of  the  first  order  and  second 
degree,  each  of  which  would  involve  one  arbitrary  quantity,  by  the  variation  of 
which  we  should  pass  from  one  curve  to  another  of  the  same  system. 

Hence  the  position  of  any  curve  of  either  system  depends  on  the  value 
assumed  for  the  arbitrary  constant ;  to  distinguish  the  systems,  let  us  call  one 
the  first  system,  and  the  other  the  second,  and  let  all  quantities  relating  to 
the  second  system  be  denoted  by  accented  letters. 

Let  the  arbitrary  constants  introduced  by  integration  be  u  for  the  first 
system,  and  u    for  the  second. 

Then  the  value  of  lo  will  determine  the  position  of  a  curve  of  the  first 
system,  and  that  of  u  a  curve  of  the  second  system,  and  therefore  u  and  u  will 
suffice  to  determine  the  point  of  intersection  of  these  two  curves. 

Hence  we  may  conceive  the  position  of  any  point  on  the  surface  to  be 
determined  by  the  values  of  u  and  u  for  the  curves  of  the  two  systems  which 
intersect  at  that  point. 

By  taking  into  account  the  equation  to  the  surface,  we  may  suppose  x,  y, 
and  2  the  co-ordinates  of  any  point,  to  be  determined  as  functions  of  the  two 
variables  u  and  u.  This  being  done,  we  shall  have  materials  for  calculating 
everything  connected  with  the  surface,  and  its  lines  of  bending.  But  before 
entering  on  such  calculations  let  us  examine  the  principal  properties  of  these  lines 
which  we  must  take  into  account. 

Suppose  a  series  of  values  to  be  given  to  u  and  u,  and  the  corresponding 
curves  to  be  drawn  on  the  surface. 

VOL,  I.  13 


98  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

The  surface  will  then  be  covered  with  a  system  of  quadrilaterals,  the  size 
of  which  may  be  diminished  indefinitely  by  interpolating  values  of  u  and  u 
between  those  already  assumed;  and  in  the  limit  each  quadrilateral  may  be 
regarded  as  a  parallelogram  coinciding  with  a  facet  of  the  inscribed  polyhedron. 

The  length,  the  breadth,  and  the  angle  of  these  parallelograms  will  vary  at 
different  parts  of  the  surface,  and  will  therefore  depend  on  the  values  of  u 
and  It. 

The  curvature  of  a  line  drawn  on  a  surface  may  be  investigated  by  consider- 
ing the  curvature  of  two  other  lines  depending  on  it. 

The  first  is  the  projection  of  the  line  on  a  tangent  plane  to  the  surface  at 
a  given  point  in  the  line.  The  curvature  of  the  projection  at  the  point  of 
contact  may  be  called  the  tangential  cwvature  of  the  line  on  the  surface.  It 
has  also  been  called  the  geodesic  curvature,  because  it  is  the  measure  of  its 
deviation  from  a  geodesic  or  shortest  line  on  the  surface. 

The  other  projection  necessary  to  define  the  curvature  of  a  line  on  the 
surface  is  on  a  plane  passing  through  the  tangent  to  the  curve  and  the  normal 
to  the  surface  at  the  point  of  contact.  The  curvature  of  this  projection  at  that 
point  may  be   called   the   normal  cw^ature   of  the   line   on   the   surface. 

It  is  easy  to  shew  that  this  normal  curvature  is  the  same  as  the  curvature 
of  a  normal  section  of  the  surface  passing  through  a  tangent  to  the  curve  at 
the  same  point. 


10.     General  considerations  applicable  to  the  inscribed  polyhedron. 

When  two  series  of  lines  of  bending  belonging  to  the  first  and  second  systems 
have  been  described  on  the  surface,  we  may  proceed,  as  in  Art.  (l),  to  describe 
a  third  series  of  curves  so  as  to  pass  through  all  their  intersections  and  form 
the  diagonals  of  the  quadrilaterals  foi-med  by  the  first  pair  of  systems. 

Plane  triangles  may  then  be  constituted  within  the  surface,  having  these 
points  of  intersection  for  angles,  and  the  size  of  the  facets  of  this  polyhedron  may 
be  diminished  indefinitely  by  increasing  the  number  of  curves  in  each  series. 

But  by  Art.  (8)  the  first  and  second  systems  of  lines  of  bending  are  conju- 
gate to  each  other,  and  therefore  by  Art.  (4)  the  polygon  just  constructed  will 
have  every  pair   of  triangular   facets   in   the   same   plane,    and  may   therefore   be 


TRANSFORMATION  OF  SURFACES  BY  BENDING.  99 

considered  as  a  polyhedron  with  plane  quadrilateral  facets  all  whose  solid  angles 
are  formed  by  four  of  these  facets  meeting  in  a  point. 

When  the  number  of  curves  in  each  system  is  increased  and  their  distance 
diminished  indefinitely,  the  plane  facets  of  the  polyhedron  will  ultimately  coincide 
with  the  curved  surface,  and  the  polygons  formed  by  the  successive  edges  between 
the  facets,  will  coincide  with  the  lines  of  bending. 

These  quadrilaterals  may  then  be  considered  as  parallelograms,  the  length 
of  which  is  determined  by  the  portion  of  a  curve  of  the  second  system  inter- 
cepted between  two  curves  of  the  first,  while  the  breadth  is  the  distance  of 
two  curves  of  the  second  system  measured  along  a  curve  of  the  first.  The 
expressions  for  these  quantities  will  be  given  when  we  come  to  the  calculation  of 
our  results  along  with  the  other  particulars  which  we  only  specify  at  present. 

The  angle  of  the  sides  of  these  parallelograms  will  be  ultimately  the  same 
as  the  angle  of  intersection  of  the  first  and  second  systems,  which  we  may 
call  <f> ;  but  if  we  suppose  the  dimensions  of  the  facets  to  be  small  quantities 
of  the  first  order,  the  angles  of  the  four  facets  which  meet  in  a  point  will  difier 
from  the  angle  of  intersection  of  the  curves  at  that  point  by  small  angles  of 
the  first  order  depending  on  the  tangential  curvature  of  the  lines  of  bending. 
The  sum  of  these  four  angles  will  differ  from  four  right  angles  by  a  small 
angle  of  the  second  order,  the  circular  measure  of  which  expresses  the  entire 
curvature  of  the  solid  angle  as  in  Art.  (2). 

The  angle  of  inclination  of  two  adjacent  facets  will  depend  on  the  normal 
curvature  of  the  lines  of  bending,  and  will  be  that  of  the  projection  of  two  con- 
secutive sides  of  the  polygon  of  one  system  on  a  plane  perpendicular  to  a  side 
of  the  other  system. 

11.     Explanation  of  the  Notation   to   be   employed  in  calculation. 

Suppose  each  system  of  lines  of  bend- 
ing to  be  determined  by  an  equation  con- 
taining one  arbitrary  parameter. 

Let  this  parameter  be  u  for  the  first 
system,  and  u'  for  the  second. 

Let  two  curves,  one  from  each  system, 
be  selected  as  curves  of  reference,  and  let 
their  parameters  be  u^  and  u\. 


100  TRANSFORMATION    OF    SURFACES*  BY    BENDING. 

Let  ON  and  OM  in  the  figure  represent  these  two  curves. 

Let  PM  be  any  curve  of  the  first  system  whose  parameter  is  u,  and  PN 
any  curve  of  the  second  whose  parameter  is  u,  then  their  intersection  P  may 
be  defined  as  the  point  (w,  u'),  and  all  quantities  referring  to  the  point  P  may 
be  expressed  as  functions  of  u  and  u. 

Let  PN,  the  length  of  a  curve  of  the  second  system  (u),  from  N  (wj  to  P 
(u),  be  expressed  by  s,  and  PM  the  length  of  the  curve  {u)  from  {u\)  to  (u),  by 
s\  then  s  and  s   will  be  functions  of  u  and  u. 

Let  (w  +  Sm)  be  the  parameter  of  the  curve  QF  of  the  first  system  consecu- 
tive to  PM.  Then  the  length  of  PQ,  the  part  of  the  curve  of  the  second  system 
intercepted  between  the  curves  (u)  and  (w  +  Sw),  will  be 

ds  ^ 
du 

Similarly  PR  may  be  expressed  by 

ds\  , 

These  values  of  PQ  and  PR  will  be  the  ultimate  values  of  the  length  and 
breadth  of  a  quadrilateral  facet. 

The  angle  between  these  lines  will  be  ultimately  equal  to  ^,  the  angle  of 
intersection  of  the  system ;  but  when  the  values  of  8w  and  hu  are  considered  as 
finite  though  small,  the  angles  a,  6,  c,  d  of  the  facets  which  form  a  soHd  angle 
will  depend  on  the  tangential  curvature  of  the  two  systems  of  lines. 

Let  T  be  the  tangential  curvature  of  a  curve  of  the  first  system  at  the 
given  point  measured  in  the  direction  in  which  u  increases,  and  let  r\  that  of  the 
second  system,  be  measured  in  the  direction  in  which  xC  increases. 

Then  we  shall  have  for  the  values  of  the  four  plane  angles  which  meet  at  P, 

,      \  ds  ^  ,      1   ds^ 

1  _,  1  c?/  ^  .      1    ds  ^ 

~^  It  du  It  du      ' 

,       \  ds  rs  ,      \    ds  ^ 
J      .  I  ds'      ,      1    ds  ^ 


TRANSFORMATION    OF    SURFACES    BY    BENDING.  101 

These  values  are  correct  as  far  as  the  first  order  of  small  quantities.  Those 
corrections  which  depend  on  the  curvature  of  the  surface  are  of  the  second  order. 

Let  p  be  the  normal  curvature  of  a  curve  of  the  first  system,  and  p  that 
of  a  curve  of  the  second,  then  the  inclination  I  of  the  plane  facets  a  and  6, 
separated  by  a  curve  of  the  second  system,  will  be 

p  sin  ^  du 
as  far  as  the  first  order  of  small  angles,  and  the  inclination  V  of  h  and  c  will  be 


7/  1  0^  ^ 

/  =  -7—. — 7  -J-  ou 
p  Bin.<f>  du 


to  the  same  order  of  exactness. 


12.     On  the  corresponding  polygon  on  the  surface  of  the  sphere  of  reference. 

By  the  method  described  in  Art.  (2)  we  may 
find  a  point  on  the  sphere  corresponding  to  each 
facet  of  the  polyhedron. 

In  the  annexed  figure,  let  a,  b,  c,  d  be  the 
points  on  the  sphere  corresponding  to  the  four  facets 
which  meet  at  the  solid  angle  P.  Then  the  area 
of  the  spherical  quadrilateral  a,  h,  c,  d  will  be  the 
measure  of  the  entire  curvature  of  the  solid  angle  P. 

This  area  is  measured  by  the  defect  of  the  sum  of  the  exterior  angles 
from  four  right  angles  ;  but  these  exterior  angles  are  equal  to  the  four  angles 
a,  h,  c,  d,  which  form  the  solid  angle  P,  therefore  the  entire  curvature  is 
measured  by 

k  =  2'rr-{a  +  h  +  c-{-d). 

Since  a,  h,  c,  d  are  invariable,  it  is  evident,  as  in  Art.  (2),  that  the  entire 
curvature  at  P  is  not  altered  by  bending. 

By  the  last  article  it  appears  that  when  the  facets  are  small  the  angles  b 
and  d  are  approximately  equal  to  <j),  and  a  and  c  to  (tt  — ^),  and  since  the  sides 
of  the  quadrilateral  on  the  sphere  are  small,  we  may  regard  it  as  approximately 
a  plane  parallelogram  whose  angle  bad  =  <f). 

The  sides  of  this  parallelogram  will  be  I  and  I',  the  supplements  of  the 
angles  of  the  edges  of  the  polyhedron,  and  we  may  therefore  express  its  area 
as  a  plane  parallelogram 

k  =  IV  sin  <f>. 


102 


TRANSFORMATION    OF    SURFACES    BY    BENDING. 


By  the  expression  for  I  and  V  in  the  last  article,  we  find 

,  1         ds  ds\    ^  , 

k  =  — r-. — 7  J-  J-/  ou  du 
pp  sm<^  du  du 

for  the  entire  curvature  of  one  solid  angle. 

Since  the  whole  number  of  solid  angles  is  equal  to  the  whole  number  of 
facets,  we  may  suppose  a  quarter  of  each  of  the  facets  of  which  it  is  composed 
to  be  assigned  to   each  solid  angle.     The  area  of  these  will  be  the  same  as  that 

of  one  whole  facet,  namely, 

,  ds  ds'  o    ^  , 
sm  9  -J-  T->  ou  ou  ; 

therefore  dividing  the   expression  for  k  by   this  quantity,  we  find  for  the  value 

of  the  specific  curvature  at  P 

1 
■^     pp  sm'<^ 
which    gives    the    specific    curvature   in   terms   of  the    normal   curvatures   of  the 
lines  of  bending  and  their  angle  of  intersection. 

13.  Further  reduction  of  this  expression  by  rmans  of  the  "  Conic  of  Con- 
tact" as  defined  in  Art.  (3). 

Let  a  and  b  be  the  semiaxes  of  the  conic  of  contact,  and  h  the  sagitta 
or  perpendicular  to  its  plane  from  the  centre  to  the  surface. 

Let  CP,  CQ  be  semidiameters  parallel  to  the 
lines  of  bending  of  the  first  and  second  systems,  and 
therefore  conjugate  to  each  other. 


By  (Art.  3), 


,  CP" 

p=^-hr 


and    p=i-j^; 
and  the  expression  for  p  in  Art.  (12),  becomes 

^~{CP.CQsm(t>)'' 

But    CP  .CQbukJ)  is   the   area   of    the    parallelogram    CPRQ,   which   is   one 

quarter    of    the    circumscribed    parallelogram,    and     therefore     by    a    well-known 

theorem 

CP  .CQsm4>  =  ah, 


TRANSFORMATION  OF  SURFACES  BY  BENDING.  103 

and  the  expression  for  p  becomes 

or  if  the  area  of  the  circumscribing  parallelogram  be  called  A, 

The  principal  radii  of  curvature  of  the  surface  are  parallel  to  the  axes  of 
the  conic  of  contact.     Let  H  and  i^  denote  these  radii,  then 

and  therefore  substituting  in  the  expression  for  p, 

1 

or  the  specific  curvature   is   the   reciprocal   of  the   product   of  the   principal   radii 
of  curvature. 

This  remarkable  expression  was  introduced  by  Gauss  in  the  memoir  referred 
to  in  a  former  part  of  this  paper.  His  method  of  investigation,  though  not 
80  elementary,  is  more  direct  than  that  here  given,  and  wUl  shew  how  this 
result  can  be  obtained  without  reference  to  the  geometrical  methods  necessary 
to  a  more  extended  inquiry  into  the  modes  of  bending. 

14.  0)1  the  variation  of  normal  curvature  of  the  lines  of  bending  as  we  pa^s 
from  one  point  of  the  surface  to  another. 

We  have  determined  the  relation  between  the  normal  curvatures  of  the 
lines  of  bending  of  the  two  systems  at  their  points  of  intersection;  we  have 
now  to  find  the  variation  of  normal  curvature  when  we  pass  from  one  hne  of 
the  first  system  to  another,  along  a  line  of  the  second. 

In  analytical  language  we  have  to  find  the  value  of 

du  \pj 

Referring  to  the  figure  in  Art.  (11),  we  shall  see  that  this  may  be  done 
if  we  can  determine  the  difierence  between  the  angle  of  inclination  of  the 
facets  a  and  h,  and  that  of  c  and  d  :  for  the  angle  I  between  a  and  b  is 

J         1        ds    5.  , 
psiJKp  du 


104  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

and  therefore  the  difference  between  the  angle  of  a  and  b  and  that  of  c  and  d  is 

~  du      ~  du  \psm<f>  du'j 
whence  the  differential  of  p  with  respect  to  u  may  be  found 

We  must  therefore  find  U,  and  this  is  done  by  means  of  the  quadrilateral 
on  the  sphere  described  in  Art.  (12). 


15.     To  find  the  values  of  hi  and  U\ 

In  the  annexed  figure  let  ahcd  repre- 
sent the  small  quadrilateral  on  the  surface 
of  the  sphere.  The  exterior  angles  a,  h, 
c,  d  are  equal  to  those  of  the  four  facets 
which  meet  at  the  point  P  of  the  surface, 
and  the  sides  represent  the  angles  which 
the  planes  of  those  facets  make  with  each 
other ;   so  that 

ah  =  l,     lc  =  l\    cd  =  l  +  U,    da  =  l'  +  Br, 

and  the  problem  is  to  determine  Bl  and  hi"  in  terms  of  the  sides  I  and  V  and 
the  angles  a,  h,  c,  d. 

On  the  sides  ha,  he  complete  the  parallelogram  ahcd. 

Produce     ad  to  p,  so  that  ap  =  aS.     Join  Bp. 
Make      eq  =  cd  and  join  dq. 
then        Bl  =  cd-  ah, 
=  cq  —  ch, 
=  -(qo  +  oB), 

Now       qo  =  qd  tan  qdo 

=  cd  sin  qcd  cot  qod, 
but  cd  =  I  nearly,  sin  qcd  =  qcd==(e  +  h-7r)  and  qod  =  <f>; 
.'.  qo^l  (c +  h- it)  cot  <f>. 


TRANSFORMATION    OF    SURFACES    BY    BENDING.  105 

Also     oS  =  -—-^ — 
Sin  bop 

=  aB  (Bap)  — — 7 
^    ^'  8m<f> 

=  l'(a+h-7T)J-r. 

Substituting  the  values  of  a,  h,  c,  d  from  Art.  (11), 
Sl=  —  (qo  +  08) 

=  —I  —,  ^-  cot  <i>Su  —  V  — T—,  - — r  Bu. 
r   du        ^  r  du   sm0 

Finally,  substituting  the  values  of  I,  V,  and  Bl  from  Art.  (14), 

d   (      \       ds"\  sj    5  ,  cot  (/)    cZs'   1    (i5  5.    ^  ,  1         ds   I   ds'       ^  , 

du  \p  sin  <p  du  /  p  sm  <f>  du  r    du  p  sm  <j>  du  r  du 

which  may  be  put  under  the  more  convenient  form 

—  n        ^  =  —  1      /    1      ^^'\      1   ds         ,      p  I   ds      1 
du^    °'^'~du     ^  \sin <j>  du)      r   du        ^     p'  r  du  sin  <^  ' 

and  from  the  value  of  Bl'  we  may  similarly  obtain 

d    ,,        '\  _  _^  1       /    1       ^\  ,i^     +^j_^i^      ^ 
du  ^    ^  ^  '     du'     °  \sin  <f>  du)     r  du'        ^     p  r   du   sin  (ft ' 

We  may  simplify  these  equations  by  putting  p   for  the  specific  curvature  of 

the  surface,  and  q  for  the  ratio     , ,  which  is  the  only  quantity  altered  by  bending. 

We  have  then 

p  =  — /   .  ,  .,      and  q  =  —,, 
^     pp  sm=<^'  ^     p 

whence     p'  =  q — ^^-r  ,  p'^  = t-tj  y 

^      ^  p  sin  <f)  9.  P  s^  Y 

and  the  equations  become 

d  ,.        \      d  ,       (     ^Tl'X      1    ds      ,  ,      2  ds      1 

In    this    way    we    may    reduce    the   problem   of    bending   a   surface   to   the 
consideration  of  one  variable  q,  by  means  of  the  lines  of  bending. 

VOL.  I.  14 


d_ 

du' 


106  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

16.     To   obtain  the  conditio  of  Instantaneous  lines  of  bending. 

We   have  now   obtained  tlie   values   of  the  differential  coefficients  of  q  with 
respect  to  each  of  the  variables  u,  u. 
From  the  equation 

we  might  find  an  equation  which  would  give  certain  conditions  of  lines  of 
bending.  These  conditions  however  would  be  equivalent  to  those  which  we  have 
already  assumed  when  we  drew  the  systems  of  lines  so  as  to  be  conjugate  to 
each  other. 

To  find  the  true  conditions  of  bending  we  must  suppose  the  form  of  the 
surface  to  vary  continuously,  so  as  to  depend  on  some  variable  t  which  we 
may  call  the  time. 

Of  the  difierent  quantities  which  enter  into  our  equations,  none  are  changed 
by  the  operation  of  bending  except  q,  so  that  in  differentiating  with  respect 
to  t  all  the  rest  may  be  considered  constant,  q  being  the  only  variable. 

Differentiating  the  equations  of  last  article  with  respect  to  t,  we  obtain 
d"     ,,        .      2  ds      1        d  ,,        . 


Whence 


c?"     ,,       .      2  ds'     1     I  d  ,.       . 


A^t'^^'^^^  = 


{.4 1- 1  si^)-'^  Tu  ^,  ii'^^H^'o^'^'  1 1  ii^^  3^.<(">^*)- 


and 


(log  l) 


dududt 


( d  /2ds      1    \      2  ds      1      d  ,        }  ^  d  ,,  2  ds      1     1     d     ,,        . 

{M?d^^^'r-di7^^d^^'^'irqdt^^'^^^ 

two    independent  values   of   the   same   quantity,    whence   the   requiied   conditions 
may  be  obtained. 


TRANSFORMATION    OF    SURFACES    BY    BENDING. 


107 


Substituting   in   these   equations   the    values   of  those   quantities  which  occur 
in  the  original  equations,  we  obtain 


I  ds  (  d  ,      , 


ds 
du 


sin 


*) 


+  -  ,  \,  cot  <!>  y 


2  ds 
r  du 


\l  ds    (  d   ,      f     ,ds     .     A      2  ds      .   ,\ 


which  is  the  condition  which  must  hold  at  every  instant  during  the  process  of 
bending  for  the  lines  about  which  the  bending  takes  place  at  that  instant. 
When  the  bending  is  such  that  the  position  of  the  lines  of  bending  on  the 
surface  alters  at  every  instant,  this  is  the  only  condition  which  is  required. 
It   is  therefore   called   the   condition   of  Instantaneous   lines   of  bending. 

17.     To  find  the  condition  of  Permanent  lines  of  bending. 

Since  q  changes  with  the  time,  the  equation  of  last  article  will  not  be 
satisfied  for  any  finite  time  unless  both  sides  are  separately  equal  to  zero.  In 
that  case  we  have  the  two  conditions 


(!)■ 


d  ,      /      ds    .     ,\      2ds      ^  ,     ^^ 
^,log(i^r^^sm<^j  +  -^,cotc^  =  0, 

y 

1  ds     ^ 
or  -  -J-  =  0. 
r  du 

|^log(i>r'^,siD<^)+|^cot<^  =  0,' 

1  d/     ^ 
or   -,  -J-,  =  0. 
r  du 

(2). 


If  the  lines  of  bending  satisfy  these  conditions,  a  finite  amount  of  bending 
may  take  place  without  changing  the  position  of  the  system  on  the  surface. 
Such  lines  are  therefore  called  Permanent  lines  of  bending. 

The  only  case  in  which  the  phenomena  of  bending  may  be  exhibited  by 
means  of  the  polyhedron  with  quadrilateral  facets  is  that  in  which  permanent 
lines  of  bending  are  chosen  as  the  boundaries  of  the  facets.  In  all  other  cases 
the  bending  takes  place  about  an  instantaneous  system  of  lines  which  is  con- 
tinually in  motion  with  respect  to  the  surface,  so  that  the  nature  of  the  poly- 
hedron would  need  to  be  altered  at  every  instant. 


14—2 


108  TRANSFORMATION  OF  SURFACES  BY  BENDING. 

We  are  now  able  to  determine  whether  any  system  of  lines  drawn  on  a 
given  surface  is  a  system  of  instantaneous  or  permanent  lines  of  bending. 

We  are  also  able,  by  the  method  of  Article  (8),  to  deduce  from  two  con- 
secutive forms  of  a  surface,  the  lines  of  bending  about  which  the  transformation 
must  have  taken  place. 

If  our  analytical  methods  were  sufficiently  powerful,  we  might  apply  our 
results  to  the  determination  of  such  systems  of  lines  on  any  known  surface,  but 
the  necessary  calculations  even  in  the  simplest  cases  are  so  compHcated,  that, 
even  if  useful  results  were  obtained,  they  would  be  out  of  place  in  a  paper  of 
this  kind,  which  is  intended  to  afford  the  means  of  forming  distinct  conceptions 
rather  than  to  exhibit  the  results  of  mathematical  labour. 

18.  On  the  application  of  the  ordinary  unethods  of  analytical  geometry  to  the 
consideration  of  lines  of  bending. 

It  may  be  interesting  to  those  who  may  hesitate  to  accept  results  derived 
from  the  consideration  of  a  polyhedron,  when  applied  to  a  curved  surface,  to 
inquire  whether  the  same  results  may  not  be  obtained  by  some  independent 
method. 

As  the  following  method  involves  only  those  operations  which  are  most 
familiar  to  the  analyst,  it  will  be  sufficient  to  give  the  rough  outline,  which  may 
be  filled  up  at  pleasure. 

The  proof  of  the  invariability  of  the  specific  curvature  may  be  taken  from 
any  of  the  memoirs  above  referred  to,  and  its  value  in  terms  of  the  equation  of 
the  surface  will  be  foimd  in  the  memoir  of  Gauss. 

Let  the  equation  to  the  surface  be  put  under  the  form 

then  the  value  of  the  specific  curvature  is 


d\  dh       d^ 


dot?  dif     dx 


~dJz'^     dz^ 
dx       dy\ 


The  definition  of  conjugate  systems  of  curves  may   be   rendered   independent 
of  the  reasoning  formerly  employed  by  the  following  modification. 


TRANSFORMATION    OF    SURFACES    BY    BENDINO.  109 

Let  a  tangent  plane  move  along  any  line  of  the  first  system,  then  if  the  line 
of  ultimate  intersection  of  this  plane  with  itself  be  always  a  tangent  to  some  line 
of  the  second  system,  the  second  system  is  said  to  be  conjugate  to  the  first. 

It  is  easy  to  show  that  the  first  system  is  also  conjugate  to  the  second. 

Let  the  system  of  curves  be  projected  on  the  plane  of  xy,  and  at  the  point 
(x,  y)  let  a  be  the  angle  which  a  projected  curve  of  the  first  system  makes  with 
the  axis  of  x,  and  /8  the  angle  which  the  projected  curve  of  the  second  system 
which  intersects  it  at  that  point  makes  with  the  same  axis.  Then  the  condition 
of  the  systems  being  conjugate  will  be  found  to  be 

a   and   y3   being   known   as   functions   of  x   and   y,    we  may  determine  the  nature 
of  the  curves  projected  on  the  plane  of  xy. 

Supposing  the  surface  to  touch  that  plane  at  the  origin,  the  length  and 
tangential  curvature  of  the  lines  on  the  surface  near  the  point  of  contact  may 
be  taken  the  same  as  those  of  their  projections  on  the  plane,  and  any  change 
of  form  of  the  surface  due  to  bending  will  not  alter  the  form  of  the  projected 
lines  indefinitely  near  the  point  of  contact.  We  may  therefore  consider  z  as  the 
only  variable  altered  by  bending;  but  in  order  to  apply  our  analysis  with  facility, 
we  may  assume 

72 

^  =  Pg  sin' a  +  PQ- sin' A 

d'z 

,    J   =  —  PQ  sin  a  cos  a  —  PQ~^  sin  y3  cos  ^, 

^  =  PQ  cos'  a  +  P^-^  cos'  /8. 

It  will  be  seen  that  these  values  satisfy  the  condition  last  given.  Near  the 
origin  we  have 


d*z  dh        d\   I*      n-   .   ,  /        n\ 


and  q=Q'*. 


110  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

Differentiating  these  values  of  -y-^ ,  &c.,  we  shall  obtain  two  values  of    ,      , 
and  of  1—7—3,  which  being  equated  will  give  two  equations  of  condition. 

Now   if  s'   be   measured  along   a   curve   of  the   first   system,    and   R  be  any 

function  of  x  and  y,  then 

dE     dR  dR  . 

-^j-y  =  -^j-  cos  a  +  -7-  sm  a, 
as       dx  ay 

,  dR  _  dR  ds' 
du'      ds  du  ' 

We  may  also  show  that  -=-^  =  - , 


,   ,,    ,  da      .       da      d  .      (ds'    .      ,\ 

and  that    cos  a  ;i —  sm  a  ;t-  =  t-  log  ( -j—,  sm  0  1 . 
cty  (j/X     cLs        \ci/U  I 


By  substituting  these  values  in  the  equations  thus  obtained,  they  are 
reduced  to  the  two  equations  given  at  the  end  of  (Art.  15).  This  method  of 
investigation  introduces  no  difficulty  except  that  of  somewhat  long  equations,  and 
is  therefore  satisfactory  as  supplementary  to  the  geometrical  method  given  at 
length. 

As  an  example  of  the  method  given  in  page  (2),  we  may  apply  it  to 
the  case  of  the  surface  whose  equation  is 


(^.)  *{rf-j-©' 


This    surface    may  be  generated   by  the  motion    of   a    straight    line    whose 
equation  is  of  the  form 


=  acosnl — j,  2/  =  asinni-f- 


t  being  the  variable,  by  the  change  of  which  we   pass    from   one    position    of  the 
line  to  another.     This  line  always  passes  through  the  circle 

z  =  0,        ar'  +  y  =  a', 
and  the  straight  lines   z  =  c,         cc=^0, 
and  z—  —c,    y  =  0, 
which  may  therefore  be  taken  as  the  directors  of  the  surface. 


TRANSFORMATION    OF    SURFACES    BY    BENDING.  Ill 

Taking  two  consecutive  positions  of  this  line,  in  which  the  values  of  t 
are  t  and  t  +  Bt,  we  may  find  by  the  ordinary  methods  the  equation  to  the 
shortest  line  between  them,  its  length,  and  the  co-ordinates  of  the  point  in  which 
it  intersects  the  first  line. 


Calling  the  length  8^, 


ac 


8C=  ,/^    Bin 2tBt, 


Ja'  +  c 
and  the  co-ordinates  of  the  point  of  intersection  are 

x  =  2a  cos' t,         y  =  2a  sin*  t,         z=  —c  cos  2t. 
The  angle  80  between  the  consecutive  lines  is 

Ja-  +  c 
The  distance  So-  between  consecutive  shortest  lines  is 

^       3a'-F-2c* 


and  the  angle  S<^  between  these  latter  lines  is 


sin  2t8t, 


'Ja'  +  c 

Hence  if  we  suppose  ^,  6,  cr,  (f),  and  t  to  vanish  together,  we  shall  have  by 
integration 

(T  =  ~—, ( 1  —  cos  2t), 

Ja'  +  c' 

By  bending  the  surface  about  its  generating  lines  we  alter  the  value  of  (ft 
in  any  manner  without  changing  4,  0,  or  or.  For  instance,  making  <^  =  0,  all  the 
generating  lines  become  parallel  to  the  same  plane.  Let  this  plane  be  that  of 
xy,  then  ^  is  the  distance  of  a  generating  line  from  that  plane.     The  projections 


o-  = 


112  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

of  the  generating  lines  on  the  plane  of  xy  will,  by  their  ultimate  intersections, 
form  a  curve,  the  length  of  which  is  measured  by  a,  and  the  angle  which  its 
tangent  makes  with  the  axis  of  x  hj  0,  6  and  o-  being  connected  by  the  equation 

^  I  1  -  cos 6  , 

which  shows  the  curve  to  be  an  epicycloid. 

The  generating  lines  of  the  surface  when  bent  into  this  form  are  therefore 
tangents  to  a  cylindrical  surface  on  an  epicycloidal  base,  touching  that  surface 
along  a  curve  which  is  always  equally  inclined  to  the  plane  of  the  base,  the 
tangents  themselves  being  drawn  parallel  to  the  base. 

We  may  now  consider  the   bending  of  the  surface  of  revolution 

Putting  r  =  Jaf  +  f,  then  the  equation  of  the  generating  line  is 

r^  +  z^  =  c^. 
This  is  the  well-known  hypocycloid  of  four  cusps. 

Let  s  be  the  length  of  the  curve  measured  from  the  cusp  in  the  axis  of  z, 
then, 

s  =  |<jV\ 
wherefore,  r  =  (|)'  c  "  *  5^. 

Let  6  be  the  angle  which  the  plane  of  any  generating  line  makes  with 
that  of  xz,  then  s  and  6  determine  the  position  of  any  point  on  the  surface. 
The  length  and  breadth  of  an  element  of  the  surface  will  be  Ss  and  rB$. 

Now  let  the  surface  be  bent  in  the  manner  formerly  described,  so  that  0 
becomes  0^,  and  r,  r,  when 

0^  =  1x0  and  r'  =  -ry 
then        r'  =  (f)'c-V"'s' 

provided        o'  =  /u,'c. 
The   equation   between  r'  and  s  being  of  the   same  form  as  that  between 
r  and  ^   shows  that  the  surface  when   bent  is  similar  to  the  original  surface,  its 
dimensions  being  multiphed  by  fi*. 


TRANSFORMATION    OF    SURFACES    BY    BENDING.  113 

This,  however,  is  true  only  for  one  half  of  the  surface  when  bent.  The 
other  half  is  precisely  symmetrical,  but  belongs  to  a  surface  which  is  not  con- 
tinuous with  the  first. 

The  surface  in  its  original  form  is  divided  by  the  plane  of  xy  into  two 
parts  which  meet  in  that  plane,  forming  a  kind  of  cuspidal  edge  of  a  circular 
form  which  limits   the  possible  value  of  s  and  r. 

After  being  bent,  the  surface  still  consists  of  the  same  two  parts,  but  the 
edge    in   which   they   meet    is   no    longer   of  the  cuspidal   form,  but   has   a   finite 

angle  =  2  cos"^  - ,   and  the  two  sheets  of  the  surface  become  parts  of  two  different 

surfaces  which  meet  but  are  not  continuous. 


NOTE. 

As  an  example  of  the  application  of  the  more  general  theory  of  "  lines  of  bending,"  let  us 
consider  the  problem  which  has  been  already  solved  by  Professor  Jellett. 

To  determine  the  conditions  under  which  one  portion  of  a  surface  may  he  rendered  rigid,  while 
the  remainder  is  flexible. 

Suppose  the  lines  of  bending  to  be  traced  on  the  surface,  and  the  corresponding  poly- 
hedron to  be  formed,  as  in  (9)  and  (10),  then  if  the  angle  of  one  of  the  four  edges  which 
meet  at  any  solid  angle  of  the  polyhedron  be  altered  by  bending,  those  of  the  other  three 
must  be  also  altered.  These  edges  terminate  in  other  solid  angles,  the  forms  of  which  will 
also  be  changed,  and  therefore  the  efifect  of  the  alteration  of  one  angle  of  the  polyhedron  will 
be  communicated  to  every  other  angle  within  the  system  of  lines  of  bending  which  defines 
the  form  of  the  polyhedron. 

If  any  portion  of  the  surface  remains  unaltered  it  must  lie  beyond  the  limits  of  the 
system  of  lines  of  bending.  We  must  therefore  investigate  the  conditions  of  such  a  system 
being  bounded. 

The  boundary  of  any  system  of  lines  on  a  surface  is  the  curve  formed  by  the  ultimate  inter- 
section of  those  lines,  and  therefore  at  any  given  point  coincides  in  direction  with  the  curve  of 
the  system  which  passes  through  that  point.  In  this  case  there  are  two  systems  of  lines  of 
bending,  which  are  necessarily  coincident  in  extent,  and  must  therefore  have  the  same  boundary. 
At  any  point  of  this  boundary  therefore  the  directions  of  the  lines  of  bending  of  the  first 
and  second  systems  are  coincident. 

But,  by  (7),  these  two  directions  must  be  "conjugate"  to  each  other,  that  is,  must  corre- 
spond to  conjugate  diameters  of  the  "Conic  of  Contact."      Now  the  only  case  in  which  con- 
VOL.  I.  15 


114  TRANSFORMATION    OF    SURFACES    BY    BENDING. 

jugate  diameters  of  a  conic  can  coincide,  is  when  the  conic  is  an  hyperbola,  and  both  diameters 
coincide  with  one  of  the  asymptotes ;  therefore  the  boundary  of  the  system  of  lines  of  bending 
must  be  a  curve  at  every  point  of  which  the  conic  of  contact  is  an  hyperbola,  one  of  whose 
asymptotes  lies  in  the  direction  of  the  curve.  The  radius  of  "  normal  curvature "  must  there- 
fore by  (3)  be  infinite  at  eveiy  point  of  the  curve.  This  is  the  geometrical  property  of 
what  Professor  Jellett  calls  a  "  Curve  of  Flexure,"  so  that  we  may  express  the  result  as 
follows  : 

If  one  portion  of  a  surface  be  fixed,  while  the  remainder  is  bent,  the  boundary  of  the  fixed 
portion  is  a  curve  of  fiexure. 

This  theorem  includes  those  given  at  p.  (92),  relative  to  a  fixed  curve  on  a  surface,  for  in 
a  surface  whose  curvatures  are  of  the  same  sign,  there  can  be  no  "curves  of  flexure,"  and 
in  a  developable  surface,  they  are  the  rectilinear  sections.  Although  the  cuspidal  edge,  or 
arete  de  rebroussement,  satisfies  the  analytical  condition  of  a  curve  of  flexure,  yet,  since  its 
form  determines  that  of  the  whole  surface,  it  cannot  remain  fixed  while  the  form  of  the  surface 
is  changed. 

In  concavo-convex  surfaces,  the  curves  of  flexure  must  either  have  tangential  curvature  or 
be  straight  lines.  Now  if  we  put  <^=0  in  the  equations  of  Art.  (17),  we  find  that  the 
lines  of  bending  of  both  systems  have  no  tangential  curvature  at  the  point  where  they  touch 
the  curve  of  flexure.  They  must  therefore  lie  entirely  on  the  convex  side  of  that  curve,  and 
therefore 

If  a  curve  of  fiexure  be  fi^ed,  the  surface  on  the  concave  side  of  the  curve  is  not  flexible. 

I  have  not  yet  been  able  to  determine  whether  the  surface  is  inflexible  on  the  convex  side 
of  the  curve.  It  certainly  is  so  in  some  cases  which  I  have  been  able  to  work  out,  but  I 
have  no  general  proof. 

When  a  surface  has  one  or  more  rectilinear  sections,  the  portions  of  the  surface  between 
them  may  revolve  as  rigid  bodies  round  those  lines  as  axes  in  any  manner,  but  no  other  motion 
is  possible.  The  case  in  which  the  rectilinear  sections  form  an  infinite  series  has  been  discussed 
in  Sect.  (I.). 


[From   the  Cambridge  and  Dublin  Mathematical  Journal,  Vol.  ix. 


V.     On  a  particular  case  of  the  descent  of  a  heavy  body  in  a  resisting 

medium. 

Every  one  must  have  observed  that  when  a  slip  of  paper  falls  through 
the  air,  its  motion,  though  undecided  and  wavering  at  first,  sometimes  becomes 
regular.  Its  general  path  is  not  in  the  vertical  direction,  but  inclined  to  it 
at  aji  angle  which  remains  nearly  constant,  and  its  fluttering  appearance  will 
be  found  to  be  due  to  a  rapid  rotation  round  a  horizontal  axis.  The  direction 
of  deviation  from  the  vertical  depends  on  the  direction  of  rotation. 

If  the  positive  directions  of  an  axis  be  toward  the  right  hand  and  upwards, 
and  the  positive  angular  direction  opposite  to  the  direction  of  motion  of  the 
hands  of  a  watch,  then,  if  the  rotation  is  in  the  positive  direction,  the  hori- 
zontal part  of  the  mean  motion  will  be  positive. 

These  efiects  are  commonly  attributed  to  some  accidental  peculiarity  in  the 
form  of  the  paper,  but  a  few  experiments  with  a  rectangular  slip  of  paper 
(about  two  inches  long  and  one  broad),  will  shew  that  the  direction  of  rotation 
is  determined,  not  by  the  irregularities  of  the  paper,  but  by  the  initial  circum- 
stances of  projection,  and  that  the  symmetry  of  the  form  of  the  paper  greatly 
increases  the  distinctness  of  the  phenomena.  We  may  therefore  assume  that 
if  the  form  of  the  body  were  accurately  that  of  a  plane  rectangle,  the  same 
effects  would  be  produced. 

The  following  investigation  is  intended  as  a  general  explanation  of  the  true 
cause  of  the  phenomenon. 

I  suppose  the  resistance  of  the  air  caused  by  the  motion  of  the  plane  to 
be  in  the  direction  of  the  normal  and  to  vary  as  the  square  of  the  velocity 
estimated  in  that  direction. 

Now  though  this  may  be  taken  as  a  sufficiently  near  approximation  to  the 
magnitude   of  the  resisting  force   on   the   plane   taken   as   a  whole,   the  pressure 

15—2 


116  DESCENT    OF    A    HEAVY    BODY    IN    A    RESISTING    MEDIUM. 

on  any  given  element  of  the  surface  will  vary  with  its  position  so  that  the 
resultant  force  will  not  generally  pass  through  the  centre  of  gravity. 

It  is  found  by  experiment  that  the  position  of  the  centre  of  pressure 
depends  on  the  tangential  part  of  the  motion,  that  it  lies  on  that  side  of  the 
centre  of  gravity  towards  which  the  tangential  motion  of  the  plane  is  directed, 
and  that  its  distance  from  that  point  increases  as  the  tangential  velocity  in- 
creases. 

I  am  not  aware  of  any  mathematical  investigation  of  this  effect.  The 
explanation  may  be  deduced  from  experiment. 

Place  a  body  similar  in  shape  to  the  sHp  of  paper  obliquely  in  a  current 
of  some  visible  fluid.  Call  the  edge  where  the  fluid  first  meets  the  plane  the 
first  edge,  and  the  edge  where  it  leaves  the  plane,  the  second  edge,  then  we 
may  observe  that 

(1)  On  the  anterior  side  of  the  plane  the  velocity  of  the  fluid  increases 
as  it  moves  along  the  surface  from  the  first  to  the  second  edge,  and  therefore 
by  a  known  law  in  hydrodynamics,  the  pressure  must  diminish  from  the  first 
to  the  second  edge. 

(2)  The  motion  of  the  fluid  behind  the  plane  is  very  unsteady,  but  may 
be  observed  to  consist  of  a  series  of  eddies  diminishing  in  rapidity  as  they 
pass  behind  the  plane  from  the  first  to  the  second  edge,  and  therefore  relieving 
the  posterior  pressure  most  at  the  first  edge. 

Both  these  causes  tend  to  make  the  total  resistance  greatest  at  the  first 
edge,  and  therefore  to  bring  the  centre  of  pressure  nearest  to  that  edge. 

Hence  the  moment  of  the  resistance  about  the  centre  of  gravity  will  always 
tend  to  turn  the  plane  towards  a  position  perpendicular  to  the  direction  of  the 
current,  or,  in  the  case  of  the  slip  of  paper,  to  the  path  of  the  body  itself.  It 
will  be  shewn  that  it  is  this  moment  that  maintains  the  rotatory  motion  of 
the  falling  paper. 

When  the  plane  has  a  motion  of  rotation,  the  resistance  will  be  modified 
on  account  of  the  unequal  velocities  of  difierent  parts  of  the  surface.  The 
magnitude  of  the  whole  resistance  at  any  instant  will  not  be  sensibly  altered 
if  the  velocity  of  any  point  due  to  angular  motion  be  small  compared  with  that 
due  to  the  motion  of  the  centre  of  gravity.  But  there  will  be  an  additional 
moment  of  the  resistance  round  the  centre  of  gravity,  which  will  always  act  in 
the  direction  opposite  to  that  of  rotation,  and  wOl  vary  directly  as  the  normal 
and  angular  velocities  together. 


DESCENT    OF    A    HEAVY    BODY     IN    A     RESISTING     MEDIUM.  117 

The  part  of  the  moment  due  to  the  obliquity  of  the  motion  will  remain 
nearly  the  same  as  before. 

We  are  now  prepared  to  give  a  general  explanation  of  the  motion  of  the 
slip  of  paper  after  it  has  become  regular. 

Let  the  angular  position  of  the  paper  be  determined  by  the  angle  between 
the  normal  to  its  surface  and  the  axis  of  x,  and  let  the  angular  motion  be 
such  that  the  normal,  at  first  coinciding  with  the  axis  of  x,  passes  towards 
that  of  y. 

The  motion,  speaking  roughly,  is  one  of  descent,  that  is,  in  the  negative 
direction  along  the  axis  of  y. 

The  resolved  part  of  the  resistance  in  the  vertical  direction  will  always 
act  upwards,  being  greatest  when  the  plane  of  the  paper  is  horizontal,  and 
vanishing  when  it  is  vertical. 

When  the  motion  has  become  regular,  the  effect  of  this  force  during  a 
whole  revolution  will  be  equal  and  opposite  to  that  of  gravity  during  the  same 
time. 

Since  the  resisting  force  increases  while  the  normal  is  in  its  first  and  third 
quadrants,  and  diminishes  when  it  is  in  its  second  and  fourth,  the  maxima  of 
velocity  will  occur  when  the  normal  is  in  its  first  and  third  quadrants,  and 
the  minima  when  it  is  in  the  second  and  fourth. 

The  resolved  part  of  the  resistance  in  the  horizontal  direction  will  act  in 
the  positive  direction  along  the  axis  of  x  in  the  first  and  third  quadrants,  and 
in  the  negative  direction  during  the  second  and  fourth;  but  since  the  resistance 
increases  with  the  velocity,  the  whole  effect  during  the  first  and  third  quadrants 
will  be  greater  than  the  whole  effect  during  the  second  and  fourth.  Hence 
the  horizontal  part  of  the  resistance  will  act  on  the  whole  in  the  positive 
direction,  and  will  therefore  cause  the  general  path  of  the  body  to  incline  in 
that  direction,  that  is,  toward  the  right. 

That  part  of  the  moment  of  the  resistance  about  the  centre  of  gravity 
which  depends  on  the  angular  velocity  will  vary  in  magnitude,  but  wUl  always 
act  in  the  negative  direction.  The  other  part,  which  depends  on  the  obliquity 
of  the  plane  of  the  paper  to  the  direction  of  motion,  will  be  positive  in  the 
first  and  third  quadrants  and  negative  in  the  second  and  fourth ;  but  as  its 
magnitude  increases  with  the  velocity,  the  positive  effect  will  be  greater  than 
the  negative. 

When    the    motion    has    become    regular,    the   effect   of   this   excess   in   the 


118  DESCENT    OF    A    HEAVY    BODY    IN    A    RESISTING    MEDIUM. 

positive  direction  will  be  equal  and  opposite  to  the  negative  effect  due  to  the 
angular   velocity  during  a  whole  revolution. 

The  motion  will  then  consist  of  a  succession  of  equal  and  similar  parts 
performed  in  the  same  manner,  each  part  corresponding  to  half  a  revolution  of 
the  paper. 

These  considerations  will  serve  to  explain  the  lateral  motion  of  the  paper, 
and  the  maintenance  of  the  rotatory  motion. 

Similar  reasoning  will  shew  that  whatever  be  the  initial  motion  of  the 
paper,  it  cannot  remain  uniform. 

Any  accidental  oscillations  will  increase  till  their  amphtude  exceeds  half  a 
revolution.  The  motion  will  then  become  one  of  rotation,  and  will  continually 
approximate  to  that  which  we  have  just  considered. 

It  may  be  also  shewn  that  this  motion  will  be  unstable  unless  it  take 
place  about  the  longer  axis  of  the  rectangle. 

If  this  axis  is  incHned  to  the  horizon,  or  if  one  end  of  the  slip  of  paper 
be  different  from  the  other,  the  path  will  not  be  straight,  but  in  the  form  of 
a  helix.  There  will  be  no  other  essential  difference  between  this  case  and  that 
of  the  symmetrical  arrangement. 

Trinity  College,  April  5,  1853. 


[From  the  Transactions  of  the  Royal  Scottish  Society  of  Arts,  Vol.  iv.  Part  in] 


VI.     On  the  Theory  of  Colours  in  relation  to  Colour-Blindness. 
A  letter  to  Dr  G.  Wilson. 

Dear  Sir, — As  you  seemed  to  think  that  the  results  which  I  have  obtained 
in  the  theory  of  colours  might  be  of  service  to  you,  I  have  endeavoured  to 
arrange  them  for  you  in  a  more  convenient  form  than  that  in  which  I  first 
obtained  them.  I  must  premise,  that  the  first  distinct  statement  of  the  theory 
of  colour  which  I  adopt,  is  to  be  found  in  Young's  Lectures  on  Natural  Philo- 
sophy (p.  345,  Kelland's  Edition) ;  and  the  most  philosophical  enquiry  into  it 
which  I  have  seen  is  that  of  Helmholtz,  which  may  be  found  in  the  Annals  of 
Philosophy  for  1852. 

It  is  well  known  that  a  ray  of  light,  from  any  source,  may  be  divided  by 
means  of  a  prism  into  a  number  of  rays  of  different  refranglbility,  forming  a 
series  called  a  spectrum.  The  intensity  of  the  light  is  different  at  different 
points  of  this  spectrum ;  and  the  law  of  intensity  for  different  refrangibilities 
differs  according  to  the  nature  of  the  incident  light.  In  Sir  John  F.  W. 
Herschel's  Treatise  on  Light,  diagrams  will  be  found,  each  of  which  represents 
completely,  by  means  of  a  curve,  the  law  of  the  intensity  and  refranglbility  of 
a  beam  of  solar  light  after  passing  through  -various  coloured  media. 

I  have  mentioned  this  mode  of  defining  and  registering  a  beam  of  light, 
because  it  is  the  perfect  expression  of  what  a  beam  of  light  is  in  itself,  con- 
sidered with  respect  to  all  its  properties  as  ascertained  by  the  most  refined 
instruments.  When  a  beam  of  light  falls  on  the  human  eye,  certain  sensations 
are  produced,  from  which  the  possessor  of  that  organ  judges  of  the  colour  and 
intensity  of  the  light.  Now,  though  every  one  experiences  these  sensations,  and 
though  they  are  the  foundation  of  all  the  phenomena  of  sight,  yet,  on  account 
of  their  absolute  simplicity,  they  are  incapable  of  analysis,  and  can  never  become 
in   themselves   objects   of  thought.      If  we   attempt   to   discover   them,    we   must 


120  THE    THEORY    OF    COLOURS    IN    RELATION    TO    COLOUR-BLINDNESS. 

do  SO  by  artificial  means ;  and  our  reasonings   on  tKem  must  be  guided  by  some 
theory. 

The  most  general  form  in  which  the  existing  theory  can  be  stated  is  this, — 
There    are    certain    sensations,    finite    in    number,    but   infinitely  variable   in 

degree,    which   may  be   excited  by  the   difierent   kinds  of  light.      The   compound 

sensation  resulting  from  all  these  is  the  object   of  consciousness,  is  a   simple  act 

of  vision. 

It  is   easy  to   see   that  the  numher  of  these  sensations  corresponds  to  what 

may  be  called  in  mathematical  language  the  number  of  independent  variables,  of 

which  sensible  colour  is  a  function. 

This  will  be  readily  understood  by  attending  to  the  following  cases : — 

1.  When  objects  are  illuminated  by  homogeneous  yellow  light,  the  only 
thing  which  can  be  distinguished  by  the  eye  is  difference  of  intensity  or 
brightness. 

If  we  take  a  horizontal  line,  and  colour  it  black  at  one  end,  with  increasing 
degrees  of  intensity  of  yellow  light  towards  the  other,  then  every  visible  object 
wiU  have  a  brightness  corresponding  to  some  point  in  this  line. 

In  this  case  there  is  nothing  to  prove  the  existence  of  more  than  one 
sensation  in  vision. 

In  those  photographic  pictures  in  which  there  is  only  one  tint  of  which 
the  different  intensities  correspond  to  the  different  degrees  of  illumination  of  the 
object,  we  have  another  illustration  of  an  optical  effect  depending  on  one  variable 
only. 

2.  Now,  suppose  that  different  kinds  of  light  are  emanating  from  different 
sources,  but  that  each  of  these  sources  gives  out  perfectly  homogeneous  light, 
then  there  will  be  two  things  on  which  the  nature  of  each  ray  will  depend : — 
(1)  its  intensity  or  brightness ;  (2)  its  hue,  which  may  be  estimated  by  its 
position  in  the  spectrum,  and  measured  by  its  wave  length. 

If  we  take  a  rectangular  plane,  and  illuminate  it  with  the  different  kinds 
of  homogeneous  light,  the  intensity  at  any  point  being  proportional  to  its  hori- 
zontal distance  along  the  plane,  and  its  wave  length  being  proportional  to  its 
height  above  the  foot  of  the  plane,  then  the  plane  will  display  every  possible 
variety  of  homogeneous  light,  and  will  furnish  an  instance  of  an  optical  effect 
depending  on  two  variables. 


THE    THEORY    OF    COLOURS    IN     RELATION    TO    COLOUR-BLINDNESS. 


121 


3.  Now,  let  us  take  the  case  of  nature.  We  find  that  colours  differ  not 
only  in  intensity  and  Ime,  but  also  in  tint ;  that  is,  they  are  more  or  less  pure. 
We  might  arrange  the  varieties  of  each  colour  along  a  line,  which  should  begin 
with  the  homogeneous  colour  as  seen  in  the  spectrum,  and  pass  through  all 
gradations  of  tint,  so  as  to  become  continually  purer,  and  terminate  in  white. 

We  have,  therefore,  three  elements  in  our  sensation  of  colour,  each  of  which 
may  vary  independently.  For  distinctness  sake  I  have  spoken  of  intensity,  hue, 
and  tint ;  but  if  any  other  three  independent  qualities  had  been  chosen,  the 
one  set  might  have  been  expressed  in  terms  of  the  other,  and  the  results  identified. 

The  theory  which  I  adopt  assumes  the  existence  of  three  elementary  sen- 
sations, by  the  combination  of  which  all  the  actual  sensations  of  colour  are 
produced.  It  will  be  shewn  that  it  is  not  necessary  to  specify  any  given  colours 
as  typical  of  these  sensations.  Young  has  called  them  red,  green,  and  violet ;  but 
any  other  three  colours  might  have  been  chosen,  provided  that  white  resulted 
from  their  combination  in  proper  proportions. 

Before  going  farther  I  would  observe,  that  the  important  part  of  the  theoiy 
is  not  that  three  elements  enter  into  our  sensation  of  colour,  but  that  there  are 
only  three.  Optically,  there  are  as  many  elements  in  the  composition  of  a  ray 
of  light  as  there  are  different  kinds  of  light  in  its  spectrum;  and,  therefore, 
strictly  speaking,  its  nature  depends  on  an  infinite  number  of  independent 
variables. 

I  now  go  on  to  the  geometrical  form  into  which  the  theory  may  be  thrown. 
Let   it   be   granted   that   the   three   pure  sensations   corre- 
spond to  the  colours  red,  green,  and  violet,   and  that  we 
can    estimate    the    intensity    of   each    of  these    sensations 
numerically. 

Let  V,  r,  g  be  the  angular  points  of  a  triangle,  and 
conceive  the  three  sensations  as  having  their  positions  at 
these  points.  If  we  find  the  numerical  measure  of  the 
red,  green,  and  violet  parts  of  the  sensation  of  a  given 
colour,  and  then  place  weights  proportional  to  these  parts 

at  r,  g,  and  v,  and  find  the  centre  of  gravity  of  the  three  weights  by  the 
ordinary  process,  that  point  will  be  the  position  of  the  given  colour,  and  the 
numerical  measure  of  its  intensity  will  be  the  sum  of  the  tliree  primitive 
sensations. 

In    this    way,    every    possible    colour    may    have    its   position   and    intensity 

VOL.  I.  16 


122  THE    THEORY    OF    COLOURS    IN    RELATION    TO    COLOUR-BLINDNESS. 

ascertained;   and  it  is  easy  to   see  that  when   two   compound  colours  are   com- 
bined, their  centre  of  gravity  is  the  position  of  the  new  colour. 

The  idea  of  this  geometrical  method  of  investigating  colours  is  to  be  found 
in  Newton's  Opticks  (Book  I.,  Part  2,  Prop.  6),  but  I  am  not  aware  that  it  has 
been  ever  employed  in  practice,  except  in  the  reduction  of  the  experiments 
which  I  have  just  made.  The  accuracy  of  the  method  depends  entirely  on  the 
truth  of  the  theory  of  three  sensations,  and  therefore  its  success  is  a  testimony 
in  favour  of  that  theory. 

Every  possible  colour  must  be  included  within  the  triangle  rgv.  White 
will  be  foimd  at  some  point,  w,  within  the  triangle.  If  lines  be  drawn  through 
w  to  any  point,  the  colour  at  that  point  will  vary  in  hue  according  to  the 
angular  position  of  the  line  drawn  to  w,  and  the  purity  of  the  tint  will  depend 
on  the  length  of  that  line. 

Though  the  homogeneous  rays  of  the  prismatic  spectrum  are  absolutely  pure 
in  themselves,  yet  they  do  not  give  rise  to  the  "pure  sensations"  of  which  we 
are  speaking.  Every  ray  of  the  spectrum  gives  rise  to  all  three  sensations, 
though  in  different  proportions ;  hence  the  position  of  the  colours  of  the  spectrum 
is  not  at  the  boundary  of  the  triangle,  but  in  some  curve  C  R  Y  G  B  V 
considerably  within  the  triangle.  The  nature  of  this  curve  is  not  yet  determined, 
but  may  form  the  subject  of  a  future  investigation  *. 

All  natural  colours  must  be  within  this  curve,  and  all  ordinary  pigments 
do  in  fact  lie  very  much  within  it.  The  experiments  on  the  colours  of  the 
spectrum  which  I  have  made  are  not  brought  to  the  same  degree  of  accuracy  as 
those  on  coloured  papers.  I  therefore  proceed  at  once  to  describe  the  mode  of 
making  those  experiments  which  I  have  found  most  simple  and  convenient. 

The   coloured   paper  is   cut  into   the  form   of  discs,   each  with  a  small  hole 

in   the   centre,   and  divided  along  a  radius,  so  as  to  admit  ^ ^ 

of  several  of  them  being  placed  on  the  same  axis,  so  that  C^^  J 

part  of  each  is  exposed.     By  slipping  one  disc  over  another,  

we   can   expose   any  given  portion  of  each   colour.      These        >^ — ~^ 
j:«^« „i J „    ^:^.^.^^   j. j.^^i.^4. ,'4.; ^v       (       <=>       ) 


discs  are   placed  on  a  little  top  or  teetotum,  consisting  of      \^ y 

a  flat  disc  of  tin-plate  and  a  vertical  axis  of  ivory.     This 

axis  passes  through  the  centre  of  the  discs,  and  the  quantity  of  each  colour  exposed 

is  measured  by  a  graduation  on  the  rim  of  the  disc,  which  is  divided  into  100  parts. 

*  [See  the  author's  Memoir  in  the  Philosophical  Transactions,   1860,  on  the  Theory  o£  Compound 
Colours,  and  on  the  relations  of  the  Colours  of  the  Spectrum.] 


THE    THEORY    OF    COLOURS    IN     RELATION    TO    COLOUR-BLINDNESS.  123 

By  spinning  the  top,  each  colour  is  presented  to  the  eye  for  a  time  pro- 
portional to  the  angle  of  the  sector  exposed,  and  I  have  found  by  independent 
experiments,  that  the  colour  produced  by  fast  spinning  is  identical  with  that 
produced  by  causing  the  light  of  the  different  colours  to  fall  on  the  retina  at 
once. 

By  properly  arranging  the  discs,  any  given  colour  may  be  imitated  and 
afterwards  registered  by  the  graduation  on  the  rim  of  the  top.  The  principal 
use  of  the  top  is  to  obtain  colour-equations.  These  are  got  by  producing,  by 
two  different  combinations  of  colours,  the  same  mixed  tint.  For  this  purpose 
there  is  another  set  of  discs,  half  the  diameter  of  the  others,  which  lie  above 
them,  and  by  which  the  second  combination  of  colours  is  formed. 

The  two  combinations  being  close  together,  may  be  accurately  compared,  and 
when  they  are  made  sensibly  identical,  the  proportions  of  the  different  colours 
in  each  is  registered,  and  the  results  equated. 

These  equations  in  the  case  of  ordinary  vision,  are  always  between  four 
colours,  not  including  black. 

From  them,  by  a  very  simple  rule,  the  different  colours  and  compounds  have 
their  places  assigned  on  the  triangle  of  colours.  The  rule  for  finding  the  position 
is  this  : — Assume  any  three  points  as  the  positions  of  your  three  standard  colours, 
whatever  they  are  ;  then  form  an  equation  between  the  three  standard  colours, 
the  given  colour  and  black,  by  arranging  these  colours  on  the  inner  and  outer 
circles  so  as  to  produce  an  identity  when  spun.  Bring  the  given  colour  to  the 
left-hand  side  of  the  equation,  and  the  three  standard  colours  to  the  right  hand, 
leaving  out  black,  then  the  position  of  the  given  colour  is  the  centre  of  gravity 
of  three  masses,  whose  weights  are  as  the  number  of  degrees  of  each  of  the 
standard  colours,  taken  positive  or  negative,  as  the  case  may  be. 

In  this  way  the  triangle  of  colours  may  be  constructed  by  scale  and  compass 
from  experiments  on  ordinary  vision.  I  now  proceed  to  state  the  results  of 
experiments  on  Colour-Blind  vision. 

If  we  find  two  combinations  of  colours  which  appear  identical  to  a  Colour- 
Blind  person,  and  mark  their  positions  on  the  triangle  of  colours,  then  the 
straight  line  passing  through  these  points  will  pass  through  all  points  corre- 
sponding to  other  colours,  which,  to  such  a  person,  appear  identical-with  the  first 
two. 

We    may    in   the    same   way   find    other   lines  passing   through   the   series   of 

IG— 2 


124  THE    THEORY    OF    COLOURS    IN    RELATION    TO    COLOUR-BLINDNESS. 

colours  wMch  appear  alike  to  the  Colour-Blind.  All  these 
lines  either  pass  through  one  point  or  are  parallel,  ac- 
cording to  the  standard  colours  which  we  have  assumed, 
and  the  other  arbitrary  assumptions  we  may  have  made. 
Knowing  this  law  of  Colour-Blind  vision,  we  may  predict 
any  number  of  equations  which  will  be  true  for  eyes 
having  this  defect. 

The  mathematical  expression  of  the  difference  between 
Colour-BUnd  and  ordinary  vision  is,  that  colour  to  the 
former  is  a  function  of  two  independent  variables,  but  to  an  ordinary  eye,  of 
three ;  and  that  the  relation  of  the  two  kinds  of  vision  is  not  arbitrary,  but 
indicates  the  absence  of  a  determinate  sensation,  depending  perhaps  upon  some 
undiscovered  structure  or  organic  arrangement,  which  forms  one-third  of  the 
apparatus  by  which  we  receive  sensations  of  colour. 

Suppose  the  absent  structure  to  be  that  which  is  brought  most  into  play 
when  red  light  falls  on  our  eyes,  then  to  the  Colour-Blind  red  light  will  be 
visible  only  so  far  as  it  affects  the  other  two  sensations,  say  of  blue  and 
green.  It  will,  therefore,  appear  to  them  much  less  bright  than  to  us,  and  will 
excite  a  sensation  not  distinguishable  from  that  of  a  bluish-green  light. 

I  cannot  at  present  recover  the  results  of  all  my  ^periments ;  but  I  recollect 
that  the  neutral  colours  for  a  Colour-Blind  person  may  be  produced  by  com- 
bining 6  degrees  of  ultramarine  with  94  of  vermiUon,  or  60  of  emerald-green 
with  40  of  ultramarine.  The  first  of  these,  I  suppose  to  represent  to  our  eyes 
the  kind  of  red  which  belongs  to  the  red  sensation.  It  excites  the  other  two 
sensations,  and  is,  therefore,  visible  to  the  Colour-BHnd,  but  it  appears  very 
dark  to  them  and  of  no  definite  colour.  I  therefore  suspect  that  one  of  the 
three  sensations  in  perfect  vision  will  be  found  to  correspond  to  a  red  of  the 
same  hue,  but  of  much  greater  purity  of  tint.  Of  the  nature  of  the  other  two, 
I  can  say  nothing  definite,  except  that  one  must  correspond  to  a  blue,  and  the 
other  to  a  green,  verging  to  yellow. 

I  hope  that  what  I  have  written  may  help  you  in  any  way  in  your 
experiments.  I  have'  put  down  many  things  simply  to  indicate  a  way  of  thinking 
about  colours  which  belongs  to  this  theory  of  triple  sensation.  We  are  indebted 
to  Newton  for  the  original  design ;  to  Young  for  the  suggestion  of  the  means 
of   working   it   out;    to    Prof.    Forbes''    for  a   scientific   history   of  its   application 

*Phil.  Mag.  1848. 


THE    THEORY    OF    COLOURS     IN     RELATION    TO    COLOUR-BLINDNESS.  125 

to  practice;  to  Helmholtz  for  a  rigorous  examination  of  the  facts  on  which  it 
rests;  and  to  Prof  Graasman  (in  the  Phil.  Mag,  for  1852),  for  an  admirable 
theoretical  exposition  of  the  subject.  The  colours  given  in  Hay's  Nomenclature 
of  Colours  are  illustrations  of  a  similar  theory  applied  to  mixtures  of  pigments, 
but  the  results  are  often  different  from  those  in  which  the  colours  are  combined 
by  the  eye  alone.  I  hope  soon  to  have  results  with  pigments  compared  with 
those  given  by  the  prismatic  spectrum,  and  then,  perhaps,  some  more  definite 
results  may  be  obtained.     Yours  truly, 

J.  C.  MAXWELL. 


Edinburgh,  4tli  Jan.  1855. 


[From  the  Transactions  of  the  Royal  Society  of  Edinburgh,  Vol  xxi.  Part  ii.] 


VII.      Experiments  on  Colour,  as  perceived  hy  the  Eye,  with  remarks  on  Colour- 
Blindness.     Communicated  by  Dr  Gregory. 

The  object  of  tbe  following  communication  is  to  describe  a  method  by 
which  every  variety  of  visible  colour  may  be  exhibited  to  the  eye  m  such  a 
form  as  to  admit  of  accurate  comparison ;  to  shew  how  experiments  so  made 
may  be  registered  numerically;  and  to  deduce  from  these  numerical  results 
certain  laws  of  vision. 

The  different  tints  are  produced  by  means  of  a  combination  of  discs  of  paper, 
painted  with  the  pigments  commonly  used  in  the  arts,  and  arranged  round  an 
axis,  so  that  a  sector  of  any  required  angular  magnitude  of  each  colour  may  be 
exposed.  "When  this  system  of  discs  is  set  in  rapid  rotation,  the  sectors  of 
the  different  colours  become  indistinguishable,  and  the  whole  appears  of  one  uni- 
form tint.  The  resultant  tints  of  two  different  combinations  of  colours  may  be 
compared  by  using  a  second  set  of  discs  of  a  smaller  si^e,  and  placing  these  over 
the  centre  of  the  first  set,  so  as  to  leave  the  outer  portion  of  the  larger  discs 
exposed.  The  resultant  tint  of  the  first  combination  will  then  appear  in  a  ring 
round  that  of  the  second,  and  may  be  very  carefully  compared  with  it. 

The  form  in  which  the  experiment  is  most  manageable  is  that  of  the  com- 
mon top.  An  axis,  of  which  the  lower  extremity  is  conical,  carries  a  circular 
plate,  which  serves  as  a  support  for  the  discs  of  coloured  paper.  The  circumfer- 
ence of  this  plate  is  divided  into  100  equal  parts,  for  the  purpose  of  ascertainmg 
the  proportions  of  the  different  colours  which  form  the  combination.  When  the 
discs  have  been  properly  arranged,  the  upper  part  of  the  axis  is  screwed  down, 
so  as  to  prevent  any  alteration  in  the  proportions  of  the  colours. 

The  instrument  used  in  the  first  series  of  experiments  (at  Cambridge,  in 
November,  1854)  was  constructed  by  myself,  with  coloured    papers    procured   from 


EXPERIMENTS    ON    COLOUR,     AS    PERCEIVED    BY    THE    EYE. 


127 


Mr  D.  R  Hay.  The  experiments  made  in  the  present  year  were  with  the 
improved  top  made  by  Mr  J.  M.  Bryson,  Edinburgh,  and  coloured  papers  pre- 
pared by  Mr  T.  Purdie,  with  the  unmixed  pigments  used  in  the  arts.  A  number 
of  Mr  Bryson's  tops,  with  Mr  Purdie's  coloured  papers  has  been  prepared,  so  as 
to  afford  different  observers  the  means  of  testing  and  comparing  results  inde- 
pendently obtained. 

The  colour  used  for  Mr  Purdie's  papers  were — 


Vermilion 

V 

Ultramarine 

U 

Emerald  Green 

EG 

Carmine    . 

C 

Prussian  Blue  . 

PB 

Brunswick  Green 

BG 

Red  Lead 

RL 

Verditer  Blue  . 

VB 

Mixture    of    Ultramarine 

Orange  Orpiment 

00 

and  Chrome 

uc 

Orange  Chrome 

OC 

Chrome  Yellow 

CY 

Gamboge 

Gam 

Pale  Chrome    . 

PC 

Ivory  Black      . 
Snow  White     . 

Bk 
SW 

White  Paper  (Pirie,  Aberdeen), 

The  colours  in  the  first  column  are  reds,  oranges,  and  yellows;  those  in 
the  second,  blues ;  and  those  in  the  third,  greens.  Vermilion,  ultramarine,  and 
emerald  green,  seem  the  best  colours  to  adopt  in  referring  the  rest  to  a  uniform 
standard.  They  are  therefore  put  at  the  head  of  the  Hst,  as  types  of  three 
convenient  divisions  of  colour,  red,  blue,  and  green. 

It  may  be  asked,  why  some  variety  of  yellow  was  not  chosen  in  place  of 
green,  which  is  commonly  placed  among  the  secondary  colours,  while  yellow 
ranks  as  a  primary?  The  reason  for  this  deviation  from  the  received  system  is, 
that  the  colours  on  the  discs  do  not  represent  primary  colours  at  all,  but  are 
simply  specimens  of  different  kinds  of  paint,  and  the  choice  of  these  was  deter- 
mined solely  by  the  power  of  forming  the  requisite  variety  of  combinations.  Now, 
if  red,  blue,  and  yellow,  had  been  adopted,  there  would  have  been  a  difficulty 
in  forming  green  by  any  compound  of  blue  and  yellow,  while  the  yellow  formed 
by  vermilion  and  emerald  green  is  tolerably  distinct.  This  will  be  more  clearly 
perceived  after  the  experiments  have  been  discussed,  by  referring  to  the  diagram. 

As  an  example  of  the  method  of  experimenting,  let  us  endeavour  to  form  a 
neutral  gray  by  the  combination  of  vermilion,  ultramarine,  and  emerald  green. 
The    most    perfect   results   are    obtained  by  two  persons  acting  in  concert.,   when 


128  EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE. 

the  operator  arranges  the  colours  and  spins  the  top,  leaving  the  eye  of  the 
observer  free  from  the  distracting  effect  of  the  bright  colours  of  the  papers  when 
at  rest. 

After  placing  discs  of  these  three  colours  on  the  circular  plate  of  the  top, 
and  smaller  discs  of  white  and  black  above  them,  the  operator  must  spin  the 
top,  and  demand  the  opinion  of  the  observer  respecting  the  relation  of  the 
outer  ring  to  the  inner  circle.  He  will  be  told  that  the  outer  circle  is  too 
red,  too  blue,  or  too  green,  as  the  case  may  be,  and  that  the  inner  one  is  too 
light  or  too  dark,  as  compared  with  the  outer.  The  arrangement  must  then  be 
changed,  so  as  to  render  the  resultant  tint  of  the  outer  and  inner  circles  more 
nearly  alike.  Sometimes  the  observer  will  see  the  inner  circle  tinted  with  the 
complementary  colour  of  the  outer  one.  In  this  case  the  operator  must  interpret 
the  observation  with  respect  to  the  outer  circle,  as  the  inner  circle  contains  only 
black  and  white. 

By  a  little  experience  the  operator  will  learn  how  to  put  his  questions,  and 
how  to  interpret  their  answers.  The  observer  should  not  look  at  the  coloured 
papers,  nor  be  told  the  proportions  of  the  colours  during  the  experiments. 
When  these  adjustments  have  been  properly  made,  the  resultant  tints  of  the 
outer  and  inner  circles  ought  to  be  perfectly  indistinguishable,  when  the  top 
has  a  sufficient  velocity  of  rotation.  The  number  of  divisions  occupied  by  the 
different  colours  must  then  be  read  off  on  the  edge  of  the  plate,  and  registered 
in  the  form  of  an  equation.  Thus,  in  the  preceding  experiment  we  have  ver- 
milion, ultramarine,  and  emerald  green  outside,  and  black  and  white  inside.  The 
numbers,  as  given  by  an  experiment  on  the  6th  March  1855,  in  dayhght  without 
sun,  are — 

•37  V  +  -27  U  +  '36  EG  =  -28  SW+-72  Bk (1). 

The  method  of  treating  these  equations  will  be  given  when  we  come  to  the 
theoretical  view  of  the  subject. 

In  this  way  we  have  formed  a  neutral  gray  by  the  combination  of  the 
three  standard  colours.  We  may  also  form  neutral  grays  of  different  intensities 
by  the  combination  of  vermilion  and  ultramarine  with  the  other  greens,  and  thus 
obtain  the  quantities  of  each  necessary  to  neutralize  a  given  quantity  of  the 
proposed  green.  By  substituting  for  each  standard  colour  in  succession  one  of  the 
colours  which  stand  under  it,  we  may  obtain  equations,  each  of  which  contains 
two  standard  colours,  and  one  of  the  remaining  colours. 


EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE.  129 

Thus,  in  the  case  of  pale  chrome,  we  have,  from  the  same  set  of  experiments, 
•34  PC  +  -55U  +  -12  EG  =  '37  SW  +  -63Bk (2). 

"We  may  also  make  experiments  in  which  the  resultiag  tint  is  not  a  neutral 
gray,  but  a  decided  colour.  Thus  we  may  combine  ultramarine,  pale  chrome,  and 
black,  so  as  to  produce  a  tint  identical  with  that  of  a  compound  of  vermilion 
and  emerald-green.  Experiments  of  this  sort  are  more  difficult,  both  from  the 
inability  of  the  observer  to  express  the  difference  which  he  detects  in  two  tints 
which  have,  perhaps,  the  same  hue  and  intensity,  but  differ  in  purity ;  and  also 
from  the  complementary  colours  which  are  produced  in  the  eye  after  gazing  too 
long  at  the  colours  to  be  compared. 

The  best  method  of  arriving  at  a  result  in  the  case  before  us,  is  to  render 
the  hue  of  the  red  and  green  combination  something  like  that  of  the  yellow,  to 
reduce  the  purity  of  the  yellow  by  the  admixture  of  blue,  and  to  diminish  its 
intensity  by  the  addition  of  black.  These  operations  must  be  repeated  and 
adjusted,  till  the  two  tints  are  not  merely  varieties  of  the  same  colour,  but 
absolutely  the  same.     An  experiment  made  5th  March  gives — 

•39  PC-I--21  U  +  -40  Bk  =  ^59  V-f41  EG (3). 

That  these  experiments  are  really  evidence  relating  to  the  constitution  of  the 
eye,  and  not  mere  comparisons  of  two  things  which  are  in  themselves  identical, 
may  be  shewn  by  observing  these  resultant  tints  through  coloured  glasses,  or  by 
using  gas-light  instead  of  day-light.  The  tints  which  before  appeared  identical 
will  now  be  manifestly  different,  and  will  require  alteration,  to  reduce  them  to 
equality. 

Thus,  in  the  case  of  carmine,  we  have  by  day-light, 

•44  C-h-22  JJ  +  'U  EG=  •I?  SW-f-^83  Bk, 
while  by  gas-light  (Edinburgh) 

•47  C-l-^08  U-1-^45  EG  =  ^25  SW-|-^75  Bk, 
which  shews  that  the  yellowing  effect  of  the  gas-light  teUs  more  on  the  white 
than  on  the  combination  of  colours.  If  we  examine  the  two  resulting  tints 
which  appeared  identical  in  experiment  (3),  observing  the  whirling  discs  througli 
a  blue  glass,  the  combination  of  yellow,  blue,  and  black,  appears  redder  than-  the 
other,  while  through  a  yellow  glass,  the  red  and  green  mixture  appears  redder. 
So  also  a  red  glass  makes  the  first  side  of  the  equation  too  dark,  and  a  green 
glass  makes  it  too  light. 

VOL.  I.  17 


130  EXPERIMENTS    ON    COLOUR,     AS    PERCEIVED    BY    THE    EYE. 

The  apparent  identity  of  the  tints  in  these  experiments  is  therefore  not  real, 
but  a  consequence  of  a  determinate  constitution  of  the  eye,  and  hence  arises 
the  importance  of  the  results,  as  indicating  the  laws  of  human  vision. 

The  first  result  which  is  worthy  of  notice  is,  that  the  equations,  as  observed 
by  different  persons  of  ordinary  vision,  agree  in  a  remarkable  manner.  If  care 
be  taken  to  secure  the  same  kind  of  light  in  all  the  experiments,  the  equations, 
as  determined  by  two  independent  observers,  will  seldom  shew  a  difference  of 
more  than  three  divisions  in  any  part  of  the  equation  containing  the  bright 
standard  colours.  As  the  duller  colours  are  less  active  in  changing  the  resultant 
tint,  their  true  proportions  cannot  be  so  well  ascertained.  The  accuracy  of  vision 
of  each  observer  may  be  tested  by  repeating  the  same  experiment  at  different 
times,  and  comparing  the  equations  so  found. 

Experiments  of  this  kind,  made  at  Cambridge  in  November  1854,  shew  that 
of  ten  observers,  the  best  were  accurate  to  within  1^  division,  and  agreed 
within  1  division  of  the  mean  of  all ;  and  the  worst  contradicted  themselves  to 
the  extent  of  6  degrees,  but  still  were  never  more  than  4  or  5  from  the  mean 
of  all  the  observations. 

We  are  thus  led  to  conclude — 

1st.  That  the  human  eye  is  capable  of  estimating  the  likeness  of  colours 
with  a  precision  which  in  some  cases  is  very  great. 

2nd.  That  the  judgment  thus  formed  is  determined,  not  by  the  real  identity 
of  the  colours,  but  by  a  cause  residing  in  the  eye  of  the  observer. 

3rd.  That  the  eyes  of  different  observers  vary  in  accuracy,  but  agree  with 
each  other  so  nearly  as  to  leave  no  doubt  that  the  law  of  colour-vision  is 
identical  for  all  ordinary  eyes. 


Investigation  of  the  Law  of  the  Perception  of  Colour. 

Before  proceeding  to  the  deduction  of  the  elementary  laws  of  the  perception 
of  colour  from  the  numerical  results  previously  obtained,  it  will  be  desirable 
to  point  out  some  general  features  of  the  experiments  which  indicate  the  form 
which  these  laws  must  assume. 

Betuming  to  experiment  (1),  in  which  a  neutral  gray  was  produced  from 
red,  blue,    and  green,    we   may   observe,    that,  while  the  adjustments  were  incom- 


EXPERIMENTS    ON    COLOUR,    AS    PERCEIVED    BY    THE    EYE.  131 

plete,  the  difference  of  the  tints  could  be  detected  only  by  one  circle  appearing 
more  red,  more  green,  or  more  blue  than  the  other,  or  by  being  lighter  or 
darker,  that  is,  having  an  excess  or  defect  of  all  the  three  colours  together. 
Hence  it  appears  that  the  nature  of  a  colour  may  be  considered  as  dependent 
on  three  things,  as,  for  instance,  redness,  blueness,  and  greenness.  This  is  con- 
firmed by  the  fact  that  any  tint  may  be  imitated  by  mixing  red,  blue,  and 
green  alone,  provided  that  tint  does  not  exceed  a  certain  brilliancy. 

Another  way  of  shewing  that  colour  depends  on  three  things  is  by  con- 
sidering how  two  tints,  say  two  lilacs,  may  differ.  In  the  first  place,  one  may 
be  lighter  or  darker  than  the  other,  that  is,  the  tints  may  differ  in  shade. 
Secondly,  one  may  be  more  blue  or  more  red  than  the  other,  that  is,  they  may 
differ  in  hue.  Thirdly,  one  may  be  more  or  less  decided  in  its  colour ;  it  may  vary 
fi*om  purity  on  the  one  hand,  to  neutrality  on  the  other.  This  is  sometimes 
expressed  by  saying  that  they  may  differ  in  tint. 

Thus,  in  shade,  hue,  and  tint,  wo  have  another  mode  of  reducing  the 
elements  of  colour  to  three.  It  will  be  shewn  that  these  two  methods  of  con- 
sidering colour  may  be  deduced  one  from  the  other,  and  are  capable  of  exact 
numerical  comparison. 


On  a  Geographical  Method  of  Exhibiting  the  Relations  of  Colours. 

The  method  which  exhibits  to  the  eye  most  clearly  the  results  of  this  theory 
of  the  three  elements  of  colour,  is  that  which  supposes  each  colour  to  be  repre- 
sented by  a  point  in  space,  whose  distances  from  three  co-ordinate  planes  are 
proportional  to  the  three  elements  of  colour.  But  as  any  method  by  which  the 
operations  are  confined  to  a  plane  is  preferable  to  one  recLuiring  space  of  three 
dimensions,  we  shall  only  consider  for  the  present  that  which  has  been  adopted 
for  convenience,  founded  on  Newton's  Circle  of  colours  and  Mayer  and  Young's 
Triangle. 

Vermilion,  ultramarine,  and  emerald-green,  being  taken  (for  convenience)  as 
standard  colours,  are  conceived  to  be  represented  by  three  points,  taken  (for  con- 
venience) at  the  angles  of  an  equilateral  triangle.  Any  colour  compounded  of 
these  three  is  to  be  represented  by  a  point  found  by  conceiving  masses  propor- 
tional to  the  several  components  of  the  colour  placed  at  their  respective  angular 
points,  and  taking  the  centre  of  gravity  of  the  three  masses.     In  this  way,  each 

17—2 


132  EXPERIMENTS    ON    COLOUR,    AS    PERCEIVED    BY    THE    EYE. 

colour  will  indicate  by  its  position  the  proportions  of  the  elements  of  which  it  is 
composed.  The  total  intensity  of  the  colour  is  to  be  measured  by  the  whole 
number  of  divisions  of  V,  U,  and  EG,  of  which  it  is  composed.  This  may  be 
indicated  by  a  number  or  coefficient  appended  to  the  name  of  the  colour,  by 
which  the  number  of  divisions  it  occupies  must  be  multiplied  to  obtain  its  mass 
in  calculating  the  results  of  new  combinations. 

This  will  be  best  explained  by  an  example  on  the  diagram  (No.  1).  We 
have,  by  experiment  (l), 

•37  Y+-27  U  +  -36  EG=  -28  SW4-  72  Bk. 

To  find  the  position  of  the  resultant  neutral  tint,  we  must  conceive  a  mass 
of  -37  at  V,  of  -27  at  U,  and  of  '36  at  EG,  and  find  the  centre  of  gravity. 
This  may  be  done  by  taking  the  line  UV,  and  dividing  it  in  the  proportion  of 
•37  to  ^27  at  the  point  a,  where 

aV  :  aU  ::  ^27   :  '37. 

Then,  joining  a  with  EG,  divide  the  joining  line  in  W  in  the  proportion  of  ^36 
to  ("37  + "27),  W  will  be  the  position  of  the  neutral  tint  required,  which  is  not 
white,  but  0*28  of  white,  diluted  with  0^72  of  black,  which  has  hardly  any  effect 
whatever,  except  in  decreasing  the  amount  of  the  other  colour.  The  total  in- 
tensity of  our  white  paper  will  be  represented  by  oi  =  3'57;  so  that,  whenever 
white  enters  into  an  equation,  the  number  of  divisions  must  be  multiplied  by 
the  coefficient  3-57  before  any  true  results  can  be  obtained. 

We  may  take,  as  the  next  example,  the  method  of  representing  the  relation 
of  pale  chrome  to  the  standard  colours  on  our  diagram,  by  making  use  of  ex- 
periment (2),  in  which  pale  chrome,  ultramarine,  and  emerald-green,  produced  a 
neutral  gray.     The  resulting  equation  was 

•33PC  +  -55U  +  -12EG  =  -37SW  +  -63Bk (2). 

In  order  to  obtain  the  total  intensity  of  white,  we  must  multiply  the 
number  of  divisions,  -37,  by  the  proper  coefficient,  which  is  3*57.  The  result  is 
1-32,  which  therefore  measures  the  total  intensity  on  both  sides  of  the  equation. 

Subtracting  the  intensity  of  •55U  +  -12EG,  or  '67  from  1-32,  we  obtain  '65 
as  the  corrected  value  of  -33  PC.  It  will  be  convenient  to  use  these  corrected 
values  of  the  different  colours,  taking  care  to  distinguish  them  by  small  initials 
instead  of  capitals. 


EXPERIMENTS    ON    COLOUR,     AS    PERCEIVED     BY    THE    EYE.  133 

Equation  (2)  then  becomes 

•65  pc  +  -55  U  +  -12  EG  =  1  -32  w. 

Hence  pc  must  be  situated  at  a  point  such  that  w  is  the  centre  of  gravity 
of  •65pc  +  -55U  +  '12EG. 

To  find  it,  we  begin  by  determining  ^  the  centre  of  gravity  of  -55  U  +  '12EG, 
then,  joining  /8w,  the  point  we  are  seeking  must  lie  at  a  certain  distance  on 
the  other  side  of  w  from  c     This  distance  may  be  found  from  the  proportion, 

•65   :  (-55 +  -12)   ::  ^  :  w  pc, 
which   determines   the   position   of  pc.     The    proper   coefficient,  by   which  the  ob- 
served vakies  of  PC  must  be  corrected,  is  ^,  or  1-97. 

We  have  thus  determined  the  position  and  coefficient  of  a  colour  by  a  single 
experiment,  in  which  it  was  made  to  produce  a  neutral  tint  along  with  two  of 
the  standard  colours.  As  this  may  be  done  with  every  possible  colour,  the 
method  is  applicable  wherever  we  can  obtain  a  disc  of  the  proposed  colour.  In 
this  way  the  diagram  (No.  l)  has  been  laid  down  from  observations  made  in 
daylight,  by  a  good  eye  of  the  ordinary  type. 

It  has  been  observed  that  experiments,  in  which  the  resultant  tint  is  neutral, 
are  more  accurate  than  those  in  which  the  resulting  tint  has  a  decided  colour, 
as  in  experiment  (3),  owing  to  the  effects  of  accidental  colours  produced  in  the 
eye  in  the  latter  case.  These  experiments,  however,  may  be  repeated  till  a 
very  good  mean  result  has  been  obtained. 

But  since  the  elements  of  every  colour  have  been  already  fixed  by  our 
previous  observations  and  calculations,  the  agreement  of  these  results  with  those 
calculated  from  the  diagram  forms  a  test  of  the  correctness  of  our  method. 

By  experiment  (No.  3),  made  at  the  same  time  with  (l)  and  (2),  we  have 
•39PC  +  -2lU  +  -40Bk  =  -59V  +  -4lEG (3). 

Now,  joining  XJ  with  pc,  and  V  with  EG,  the  only  common  point  is  that 
at  which  they  cross,  namely  y. 

Measuring  the  parts  of  the  line  V  EG,  we  find  them  in  the  proportion  of 
•58  V  and  "42  EG  =  1*00  7. 

Similarly,  the  line  U  pc  is  divided  in  the  proportion 
78  pc  and   •22U=r00y. 


134 


EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE. 


But    -78  pc   must  be   divided  by    1-97,   to  reduce    it    to    PC,   as    was    previously 
explained.     The  result  of  calculation  is,  therefore, 

•39  PC  +  -22  U  +  -39  Bk  =  -58  V  +  "42  EG, 
the  black  being  introduced  simply  to  fill  up  the  circle. 

This  result  differs  very  little  from  that  of  experiment  (3),  and  it  must  be 
recollected  that  these  are  single  experiments,  made  independently  of  theory,  and 
chosen  at  random. 

Experiments  made  at  Cambridge,  with  all  the  combinations  of  five  colours, 
shew  that  theory  agrees  with  calculation  always  within  0-012  of  the  whole, 
and  sometimes  within  0*002.  By  the  repetition  of  these  experiments  at  the 
numerous  opportunities  which  present  themselves,  the  accuracy  of  the  results 
may  be  rendered  still  greater.  As  it  is,  I  am  not  aware  that  the  judgments 
of  the  human  eye  with  respect  to  colour  have  been  supposed  capable  of  so 
severe  a  test. 


Further  consideration  of  the  Diagram  of  Colours. 

We  have  seen  how  the  composition  of  any  tint,  in  terms  of  our  three 
standard  colours,  determines  its  position  on  the  diagram  and  its  proper  coefficient. 
In  the  same  way,  the  result  of  mixing  any  other  colours,  situated  at  other 
points  of  the  diagram,  is  to  be  found  by  taking  the  centre  of  gravity  of  their 
reduced  masses,  as  was  done  in  the  last  calculation  (experiment  3). 

We  have  now  to  turn  our  attention  to  the  general  aspect  of  the  diagram. 

The  standard  colours,  V,  U,  and  EG,  occupy  the  angles  of  an  equilateral 
triangle,  and  the  rest  are  arranged  in  the  order  in  which  they  participate  in 
red,  blue,  and  green,  the  neutral  tint  being  at  the  point  w  within  the  triangle. 
If  we  now  draw  lines  through  w  to  the  different  colours  ranged  round  it,  we 
shall  find  that,  if  we  pass  from  one  line  to  another  in  the  order  in  which  they 
lie  from  red  to  green,  and  through  blue  back  again  to  red,  the  order  will  be — 


Carmine    . 
Vermilion . 
Red  Lead . 
Oi-ange  Orpiment 
Orange  Chrome 
Chrome  Yellow 
Gramboge  . 


Coefficient. 

0-4 

Pale  Chrome 

1-0 

Mixed  Green  (U  C) 

1-3 

Brunswick  Green 

10 

Emerald  Green    . 

1-6 

Verditer  Blue     . 

1-5 

Prussian  Blue     . 

1-8 

Ultramarine 

Coefficient. 
2  0 
0-4 
0-2 
10 
0-8 
01 
10 


EXPERIMENTS    ON    COLOUR,     AS    PERCEITED    BY    THE    EYE.  135 

It  may  be  easily  seen  that  this  arrangement  of  the  colours  corresponds  to 
that  of  the  prismatic  spectrum  ;  the  only  difference  being  that  the  spectrum 
is  deficient  in  those  fine  purples  which  lie  between  ultramarine  and  vermilion, 
and  which  are  easily  produced  by  mixture.  The  experiments  necessary  for  deter- 
mining the  exact  relation  of  this  list  to  the  lines  in  the  spectrum  are  not  yet 
completed. 

If  we  examine  the  colours  represented  by  different  points  in  one  of  these 
lines  through  w,  we  shall  find  the  purest  and  most  decided  colours  at  its  outer 
extremity,  and  the  faint  tints  approaching  to  neutrality  nearer  to  w. 

If  we  also  study  the  coefficients  attached  to  each  colour,  we  shall  find  that 
the  brighter  and  more  luminous  colours  have  higher  numbers  for  their  coefficients 
than  those  which  are  dark. 

In  this  way,  the  qualities  which  we  have  already  distinguished  as  hue,  tint, 
and  shade,  are  represented  on  the  diagram  by  angular  position  with  respect  to  ir, 
distance  from  w,  and  coefficient;  and  the  relation  between  the  two  methods  of 
reducing  the  elements  of  colour  to  three  becomes  a  matter  of  geometry. 


Theory  of  the  Perception  of  Colour. 

Opticians  have  long  been  divided  on  this  point ;  those  who  trusted  to 
popular  notions  and  their  own  impressions  adopting  some  theory  of  three  primary 
colours,  while  those  who  studied  the  phenomena  of  light  itself  proved  that  no 
such  theory  could  explain  the  constitution  of  the  spectrum.  Newton,  who  was 
the  first  to  demonstrate  the  actual  existence  of  a  series  of  kinds  of  light, 
countless  in  number,  yet  all  perfectly  distinct,  was  also  the  first  to  propound 
a  method  of  calculating  the  effect  of  the  mixture  of  various  coloured  light ; 
and  this  method  was  substantially  the  same  as  that  which  we  have  just 
verified.  It  is  true,  that  the  directions  which  he  gives  for  the  construction 
of  his  circle  of  colours  are  somewhat  arbitrary,  being  probably  only  intended 
as  an  indication  of  the  general  nature  of  the  method,  but  the  method  itself 
is  mathematically  reducible  to  the  theory  of  three  elements  of  the  colour- 
sensation*. 

♦  See  Note  III.     For  a  confirmation  of  Newton's  analysis  of  Light,   see  Helmholtz,    Pogg.     Ann, 
1852;   and  Phil.  Mag.   1852,  Part  ii. 


136  EXPERIMENTS    ON    COLOUR,     AS    PERCEIVED    BY    THE    EYE. 

Youno",  who  made  the  next  great  step  in  the  establishment  of  the  theory 
of  light,  seems  also  to  have  been  the  first  to  follow  out  the  necessary  conse- 
quences of  Newton's  suggestion  on  the  mixture  of  colours.  He  saw  that,  since 
this  tripUcity  has  no  foundation  in  the  theory  of  light,  its  cause  must  be  looked 
for  in  the  constitution  of  the  eye;  and,  by  one  of  those  bold  assumptions 
which  sometimes  express  the  result  of  speculation  better  than  any  cautious 
trains  of  reasoning,  he  attributed  it  to  the  existence  of  three  distinct  modes 
of  sensation  in  the  retina,  each  of  which  he  supposed  to  be  produced  in  different 
deo-rees  by  the  different  rays.  These  three  elementary  effects,  according  to  his 
view,  correspond  to  the  three  sensations  of  red,  green,  and  violet,  and  would 
separately  convey  to  the  sensorium  the  sensation  of  a  red,  a  green,  and  a  violet 
picture ;  so  that  by  the  superposition  of  these  pictures,  the  actual  variegated 
world  is  represented*. 

In  order  fully  to  understand  Young's  theory,  the  function  which  he 
attributes  to  each  system  of  nerves  must  be  carefully  borne  in  mind.  Each  nerve 
acts,  not,  as  some  have  thought,  by  conveying  to  the  mind  the  knowledge  of  the 
length  of  an  undulation  of  light,  or  of  its  periodic  time,  but  simply  by  being 
Quore  or  less  affected  by  the  rays  which  fall  on  it.  The  sensation  of  each 
elementary  nerve  is  capable  only  of  increase  and  diminution,  and  of  no  other 
change.  We  must  also  observe,  that  the  nerves  corresponding  to  the  red 
sensation  are  affected  chiefly  by  the  red  rays,  but  in  some  degree  also  by  those 
of  every  other  part  of  the  spectrum  ;  just  as  red  glass  transmits  red  rays  freely, 
but  also  suffers  those  of  other  colours  to  pass  in  smaller  quantity. 

This  theory  of  colour  may  be  illustrated  by  a  supposed  case  taken  from 
the  art  of  photography.  Let  it  be  required  to  ascertain  the  colours  of  a  land- 
scape, by  means  of  impressions  taken  on  a  preparation  equally  sensitive  to  rays  of 
every  colour. 

Let  a  plate  of  red  glass  be  placed  before  the  camera,  and  an  impression 
taken.  The  positive  of  this  will  be  transparent  wherever  the  red  light  has  been 
abundant  in  the  landscape,  and  opaque  where  it  has  been  wanting.  Let  it  now 
be  put  in  a  magic  lantern,  along  with  the  red  glass,  and  a  red  picture  will  be 
thrown  on  the  screen. 

Let   this   operation   be   repeated   with    a    green    and    a  violet  glass,   and,  by 

*  Young's  Lectures,  p.  345,  Kelland's  Edition.  See  also  Helmholtz's  statement  of  Young's  Theory, 
in  his  Paper  referred  to  in  Note  I. ;   and  Herschel's  LigJU,  Art.  518. 


EXPERIMENTS     ON    COLOUR,     AS    PERCEIVED    BY    THE    EYE.  137 

means  of  three  magic  lanterns,  let  the  three  images  be  superimposed  on  the 
screen.  The  colour  of  any  point  on  the  screen  will  then  depend  on  that  of  the 
corresponding  point  of  the  landscape;  and,  by  properly  adjusting  the  intensities 
of  the  lights,  &c.,  a  complete  copy  of  the  landscape,  as  far  as  visible  colour  is 
concerned,  will  be  thrown  on  the  screen.  The  only  apparent  difference  will  be, 
that  the  copy  will  be  more  subdued,  or  less  pure  in  tint,  than  the  original. 
Here,  however,  we  have  the  process  performed  twice — first  on  the  screen,  and 
then  on  the  retina. 

This  illustration  will  shew  how  the  functions  which  Young  attributes  to  the 
three  systems  of  nerves  may  be  imitated  by  optical  apparatus.  It  is  therefore 
unnecessary  to  search  for  any  direct  connection  between  the  lengths  of  the 
undulations  of  the  various  rays  of  light  and  the  sensations  as  felt  by  us,  as 
the  threefold  partition  of  the  properties  of  light  may  be  effected  by  physical 
means.  The  remarkable  correspondence  between  the  results  of  experiments  on 
different  individuals  would  indicate  some  anatomical  contrivance  identical  in  all. 
As  there  is  little  hope  of  detecting  it  by  dissection,  we  may  be  content  at 
present  with  any  subsidary  evidence  which  we  may  possess.  Such  evidence  is 
furnished  by  those  individuals  who  have  the  defect  of  vision  which  was 
described  by  Dalton,  and  which  is  a  variety  of  that  which  Dr  G.  Wilson  has 
lately  investigated,   under  the   name   of  Colour-Blindness. 


Testimony  of  the  Colour- Blind  with  respect  to  Colour. 

Dr  George  Wilson  has  described  a  great  number  of  cases  of  colour- 
bhndness,  some  of  which  involve  a  general  indistinctness  in  the  appreciation 
of  colour,  while  in  others,  the  errors  of  judgment  are  plainly  more  numerous 
in  those  colours  which  approach  to  red  and  green,  than  among  those  which 
approach  to  blue  and  yellow.  In  these  more  definite  cases  of  colour-blindness, 
the  phenomena  can  be  tolerably  well  accoimted  for  by  the  hypothesis  of  an 
insensibility  to  red  light;  and  this  is,  to  a  certain  extent,  confirmed  by  the 
fact,  that  red  objects  appear  to  these  eyes  decidedly  more  obscure  than  to 
ordinary  eyes.  But  by  experiments  made  with  the  pure  spectrum,  it  appears 
that  though  the  red  appears  much  more  obscure  than  other  colours,  it  is  not 
wholly  invisible,  and,  what  is  more  curious,  resembles  the  green  more  than 
any   other   colour.     The   spectrum   to   them   appears    faintly   luminous  in  the  red; 

VOL.  L  18 


138  EXPERIMENTS    ON    COLOUR,    AS    PERCEIVED    BY    THE    EYE. 

bright  yellow  from  orange  to  yellow,  bright  but  not  coloured  from  yellow- 
green  to  blue,  and  then  strongly  coloured  in  the  extreme  blue  and  violet, 
after  which  it  seems  to  approach  the  neutral  obscure  tint  of  the  red.  It  is 
not  easy  to  see  why  an  insensibihty  to  red  rays  should  deprive  the  green 
rays,  which  have  no  optical  connection  with  them,  of  their  distinctive  appearance. 
The  phenomena  seem  rather  to  lead  to  the  conclusion  that  it  is  the  red 
serisation  which  is  wanting,  that  is,  that  supposed  system  of  nerves  which  is 
affected  in  various  degrees  by  all  light,  but  chiefly  by  red.  We  have  fortunately 
the   means  of  testing   this   hypothesis  by  numerical  results. 

Of  the  subjects  of  my  experiments  at  Cambridge,  four  were  decided  cases 
of  colour-bHndness.  Of  these  two,  namely,  Mr  E.  and  Mr  S.,  were  not 
suflficiently  critical  in  their  observations  to  afford  any  results  consistent  within 
10  divisions  of  the  colour-top.  The  remaining  two,  Mr  N.  and  Mr  X.,  were 
as  consistent  in  their  observations  as  any  persons  of  ordinary  vision  can  be, 
while  the  results  shewed  all  the  more  clearly  how  completely  their  sensations 
must  differ  from  ours. 

The  method  of  experimenting  was  the  same  as  that  adopted  with  ordinary 
eyes,  except  that  in  these  cases  the  operator  can  hardly  influence  the  result 
by  yielding  to  his  own  impressions,  as  he  has  no  perception  whatever  of  the 
similarity  of  the  two  tints  as  seen  by  the  observer.  The  questions  which  he 
must  ask  are  two,  Which  circle  appears  most  blue  or  yellow  ?  Which  appears 
lightest  and  which  darkest  ?  By  means  of  the  answers  to  these  questions  he 
must  adjust  the  resulting  tints  to  equality  in  these  respects  as  it  appears  to 
the  observer,  and  then  ascertain  that  these  tints  now  present  no  difference  of 
colour  whatever  to  his  eyes.  The  equations  thus  obtained  do  not  require  five 
colours  including  black,  but  four  only.  For  instance,  the  mean  of  several  obser- 
vations  gives — 

•19  G+'05  B  +  -76  Bk=100R (4). 

[In  these  experiments  R,  B,  G,  Y,  stand  for  red,  blue,  green,  and  yellow 
papers  prepared  by  Mr  D.  R.  Hay.  I  am  not  certain  that  they  are  identical 
with  his  standard  colours,  but  I  beUeve  so.  Their  relation  to  vermihon,  ultra- 
marine, and  emerald-green  is  given  in  diagram  (1).  Their  relations  to  each  other 
are  very  accurately  given  in  diagram  (2).] 

It  appears,  then,  that  the  dark  blue-green  of  the  left  side  of  the  equation 
is  equivalent  to  the  full  red  of  the  right  side. 


EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE.  139 

Hence,  if  we  divide  the  line  BG  in  the  proportion  19  to  5  at  the  point  y8, 
and  join  R)8,  the  tint  at  ^  will  differ  from  that  at  R  (to  the  colour-blind) 
only  in  being  more  brilliant  in  the  proportion  of  100  to  24,  and  all  inter- 
mediate tints  on  the  line  R^  will  appear  to  them  of  the  same  hue,  but 
of  intermediate   intensities. 

Now,  if  we  take  a  point  D,  so  that  RD  is  to  R^  in  the  proportion  of 
24  to  100  —  24,  or  76,  the  tint  of  D,  if  producible,  should  be  invisible  to 
the  colour-blind.  D,  therefore,  represents  the  pure  sensation  which  is  unknown 
to  the  colour-blind,  and  the  addition  of  this  sensation  to  any  others  cannot 
alter   it   in   their  estimation.     It  is  for  them  equivalent   to   black. 

Hence,  if  we  draw  lines  through  D  in  different  directions,  the  colours 
belonging  to  any  line  ought  to  differ  only  in  intensity  as  seen  by  them,  so 
that  one  of  them  may  be  reduced  to  the  other  by  the  addition  of  black 
only.  If  we  draw  DW  and  produce  it,  all  colours  on  the  upper  side  of  DW 
will  be  varieties  of  blue,  and  those  on  the  under  side  varieties  of  yellow,  so 
that  the  line  DW  is  a  boundary  line  between  their  two  kinds  of  colour,  blue 
and  yellow  being  the  names  by  which  they  call  them. 

The  accuracy  of  this  theory  will  be  evident  from  the  comparison  of  the 
experiments  which  I  had  an  opportunity  of  making  on  Mr  N.  and  Mr  X.  with 
each  other,  and  with  measurements  taken  from  the  diagram  No.  2,  which  was 
constructed  from  the  observations  of  ordinary  eyes  only,  the  point  D  alone 
being   ascertained  from  a  series  of  observations  by  Mr  N. 

Taking  the  point  y,  between  R  and  B,  it  appears,  by  measurement  of  the 
lines  Ry   and  By,   that  y  corresponds  to 

•07  B  +  -93R. 

By  measurement  of  Wy  and  Dy,  and  correction  by  means  of  the  coeflScient 
of  W,   and  caUing   D  black   in  the  colour-blind  language,  y  corresponds  to 

•105  W-f895  Bk. 
Therefore 

By  measurement    -93  R+ '07  B  =  ^105  W  + •sgs  Bk  1 

By  observation  N.  &  X.  together    "94  R-f -06  B  =  •lO    W-f-^90    Bk    I (5). 

By  X.  alone   -93  R-h-07  B  =  -10    W  +  -90    Bk  J 

The  agreement  here  is  as  near  as  can  be  expected. 

18—2 


140 


EXPERIMENTS    ON    COLOira,    AS    PERCEIVED    BY    THE    EYE. 


By  a  similar  calculation  with  respect  to  the  point  8,  between  B  and  G, 

By  measurement    -43  B  +  -57  G  =  -335  W  +  *665  Bk  1 

Observed  by  N.  and  X '41  B  +  '59  G  = '34    W  +  -66    Bk   I (6). 

By  X.  alone  -42  B  +  -58  G  =  -32    W  +  -68    Bk  J 

We  may  also  observe,  that  the  line  GD  crosses  RY.     At  the  point  of  inter- 
section we  have — 

By  calculation '87  B  +  'IS  Y  =  -34  G  +  -66  Bk 

Observed  by  N.  and  X -86  R  +  -14  Y  =  -40  G  +  'GO  Bk 

X •84R  +  '16  Y=-31  G  +  '69  Bk 

X -QOR  +  'IO  Y  =  -27  G  +  73Bk 


.(7). 


Here  observations  are  at  variance,  owing  to  the  decided  colours  produced 
affecting  the  state  of  the  retina,  but  the  mean  agrees  well  with  calculation. 

Drawing  the  line  BY,  we  find  that  it  cuts  lines  through  D  drawn  to  every 
colour.  Hence  all  colours  appear  to  the  colour-blind  as  if  composed  of  blue 
and  yellow.     By  measurement  on  the  diagram,  we  find  for  red 

Measured    -138  Y+-123  B  +  749  Bk  =  100  R' 

Observed  by  N..., -15    Y  +  'll    B-1--74    Bk  =  100RJ- (8). 

X....-13    Y  +  'll    B  +  -76    Bk  =  100R 


.(9). 


For  green  we  have  in  the  same  way — 

Measured     705  Y  +  -295  B  =  '95  G  +  -05  Bkl 

Observed  by  N....  70    Y  +  -30    B  =  -86  G  +  -14  Bk  i .... 
X....  70    Y+-30    B  =  '83  G+-17BkJ 

For  white — 

Measured     '407  Y  +  -593  B  =  '326  W  +  "674  Bk 

Observed  by  N....  -40    Y+-60    B  =  -33    W+-67    Bk 
X....  -44    Y+-56    B=-33    W+-67    Bk 

The  accuracy  of  these  results  shews  that,  whether  the  hypothesis  of  the 
want  of  one  element  out  of  three  necessary  to  perfect  vision  be  actually  true 
or  not,  it  affords  a  most  trustworthy  foundation  on  which  to  build  a  theory 
of  colour-blindness,  as  it  expresses  completely  the  observed  facts  of  the  case. 
They  also  furnish  us  with  a  datum  for  our  theory  of  perfect  vision,  namely, 
the  point  D,  which  points  out  the  exact  nature  of  the  colour-sensation,  which 
must  be   added   to   the   colour-blind   eye   to   render   it   perfect.     I  am  not  aware 


EXPERIMENTS    ON    COLOUR,     AS    PERCEIVED    BY    THE    EYE.  141 

of  any  method  of  determining  by  a  legitimate  process  the  nature  of  the  other 
two  sensations,  although  Young's  reasons  for  adopting  something  like  green  and 
violet  appear  to  me  worthy  of  attention. 

The  only  remaining  subject  to  which  I  would  call  the  attention  of  the 
Society  is  the  effect  of  coloured  glasses  on  the  colour-blind.  Although  they  can- 
not distinguish  reds  and  greens  from  varieties  of  gray,  the  transparency  of  red 
and  green  glasses  for  those  kinds  of  light  is  very  different.  Hence,  after  finding 
a  case  such  as  that  in  equation  (4),  in  which  a  red  and  a  green  appear  iden- 
tical, on  looking  through  a  red  glass  they  see  the  red  clearly  and  the  green 
obscurely,  while  through  a  green  glass  the  red  appears  dark  and  the  green  light. 

By  furnishing  Mr  X.  with  a  red  and  a  green  glass,  which  he  could  dis- 
tinguish only  by  their  shape,  I  enabled  him  to  make  judgments  in  previously 
doubtful  cases  of  colour  with  perfect  certainty.  I  have  since  had  a  pair  of 
spectacles  constructed  with  one  eye-glass  red  and  the  other  greeiL  These  Mr  X. 
intends  to  use  for  a  length  of  time,  and  he  hopes  to  acquire  the  habit  of  discri- 
minating red  from  green  tints  by  their  different  effects  on  his  two  eyes.  Though 
he  can  never  acquire  our  sensation  of  red,  he  may  then  discern  for  himself  what 
things  are  red,  and  the  mental  process  may  become  so  familiar  to  him  as  to  act 
unconsciously  like  a  new  sense. 

In  one  experiment,  after  looking  at  a  bright  light,  with  a  red  glass  over  one 
eye  and  a  green  over  the  other,  the  two  tints  in  experiment  (4)  appeared  to  him 
altered,  so  that  the  outer  circle  was  lighter  according  to  one  eye,  and  the  inner 
according  to  the  other.  As  far  as  I  could  ascertain,  it  appeared  as  if  the  eye 
which  had  used  the  red  glass  saw  the  red  circle  brightest.  This  result,  which 
seems  at  variance  with  what  might  be  expected,  I  have  had  no  opportunity  of 
verifying. 

This  paper  is  already  longer  than  was  originally  intended  For  further 
information  I  would  refer  the  reader  to  Newton's  Optich,  Book  i.  Part  ii.,  to 
Young's  Lectures  on  Natural  Philosophy,  page  345,  to  Mr  D.  R.  Hay's  works  on 
Colours,  and  to  Professor  Forbes  on  the  "Classification  of  Colours"  (Phil.  Mag., 
March,  1849). 

The  most  remarkable  paper  on  the  subject  is  that  of  M.  Helmholtz,  in  the 
Philosophical  Magazine  for  1852,  in  which  he  discusses  the  different  theories  of 
primary  colours,  and  describes  his  method  of  mixing  the  colours  of  the  spectrum. 
An    examination    of  the    results    of  M.    Helmholtz    with    reference  to   the  theory 


142  EXPERIMENTS  OX  COLOUR,  AS  PERCBIV^ED  BY  THE  EYE. 

of  three  elements  of  colour,  by  Professor  Grassmann,  is  translated  in  the  Phil. 
Mag.,  April,   1854. 

References  to  authors  on  colour-blindness  are  given  in  Dr  G.  Wilson's  papers 
on  that  subject.  A  valuable  Letter  of  Sir  J.  F.  W.  Herschel  to  Dalton  on  his 
peculiarity  of  vision,  is  to  be  found  in  the  Life  of  Dalton  by  Dr  Henry. 

I  had  intended  to  describe  some  experiments  on  the  propriety  of  the  method 
of  mixino-  colours  by  rotation,  which  might  serve  as  an  extension  of  Mr  Swan's 
experiments  on  instantaneous  impressions  on  the  eye.  These,  together  with  the 
explanation  of  some  phenomena  which  seem  to  be  at  variance  with  the  theory  of 
vision  here  adopted,  must  be  deferred  for  the  present.  On  some  future  occasion, 
I  hope  to  be  able  to  connect  these  simple  experiments  on  the  colours  of  pigments 
with  others  in  which  the  pure  hues  of  the  spectrum  are  used.  I  have  already 
constructed  a  model  of  apparatus  for  this  purpose,  and  the  results  obtained  are 
sufficiently  remarkable  to  encourage  perseverance. 


Note  I. 
On  different  Methods  of  Exhibiting  the  Mixtures  of  Colours. 

(1)  Mechanical  Mixture  of  Coloured  Powders. 
By  grinding  coloured  powders  together,  the  differently- coloured  particles  may 
be  so  intermingled  that  the  eye  cannot  distinguish  the  colours  of  the  separate 
powders,  but  receives  the  impression  of  a  uniform  tint,  depending  on  the  nature 
and  proportions  of  the  pigments  used.  In  this  way,  Newton  mixed  the  powders 
of  orpiment,  purple,  bise,  and  viride  ceris,  so  as  to  form  a  gray,  which,  in  sun- 
light, resembled  white  paper  in  the  shade.  (Newton's  Opticks,  Book  i.  Part  n., 
Exp.  XV.)  This  method  of  mixture,  besides  being  adopted  by  all  painters,  has 
been  employed  by  optical  writers  as  a  means  of  obtaining  numerical  results. 
The  specimens  of  such  mixtures  given  by  B.  R.  Hay  in  his  works  on  Colour, 
and  the  experiments  of  Professor  J.  D.  Forbes  on  the  same  subject,  shew  the 
importance  of  the  method  as  a  means  of  classifying  colours.  There  are  two 
objections,  however,  to  this  method  of  exhibiting  colours  to  the  eye.  When 
two  powders  of  unequal  fineness  are  mixed,  the  particles  of  the  finer  powder 
cover  over  those  of  the  coarser,  so  as  to  produce  more  than  their  due  effect 
in   influencing   the   resultant  tint.     For  instance,  a  small  quantity  of  lamp-black. 


EXPERIMENTS    ON    COLOUR,     AS    PERCEIVED     BY    THE     EYE.  143 

mixed  with  a  large  quantity  of  chalk,  will  produce  a  mixture  which  is  nearly 
black.  Although  the  powders  generally  used  are  not  so  different  in  this  respect 
as  lamp-black  and  chalk,  the  results  of  mixing  given  weights  of  any  coloured 
powders  must  be  greatly  modified  by  the  mode  in  which  these  powders  have 
been  prepared. 

Again,  the  light  which  reaches  the  eye  from  the  surface  of  the  mixed  pow- 
ders consists  partly  of  light  which  has  fallen  on  one  of  the  substances  mixed 
without  being  modified  by  the  other,  and  partly  of  light  which,  by  repeated 
reflection  or  transmission,  has  been  acted  on  by  both  substances.  The  colour  of 
these  rays  will  not  be  a  mixture  of  those  of  the  substances,  but  will  be  the 
result  of  the  absorption  due  to  both  substances  successively.  Thus,  a  mixture  of 
yellow  and  blue  produces  a  neutral  tint  tending  towards  red,  but  the  remainder 
of  white  light,  after  passing  through  both,  is  green;  and  this  green  is  generally 
sufficiently  powerful  to  overpower  the  reddish  gray  due  to  the  separate  colours 
of  the  substances  mixed.  This  curious  result  has  been  ably  investigated  by 
Professor  Helmholtz  of  Konigsberg,  in  his  Memoir  on  the  Theory  of  Compound 
Colours,  a  translation  of  which  may  be  found  in  the  Annals  of  Philosophy  for 
1852,  Part  2. 

(2)     Mixture  of  differently-coloured  Beams  of  Light   by   Superposition 

on  an  Opaque  Screen. 
When  we  can  obtain  light  of  sufficient  intensity,  this  method  produces  the 
most  beautiful  results.  The  best  series  of  experiments  of  this  kind  are  to  be 
found  in  Newton's  Opticks,  Book  i.  Part  ii.  The  different  arrangements  for 
mixing  the  rays  of  the  spectrum  on  a  screen,  as  described  by  Newton,  form 
a  very  complete  system  of  combinations  of  lenses  and  prisms,  by  which  almost 
every  possible  modification  of  coloured  light  may  be  produced.  The  principal 
objections  to  the  use  of  this  method  are— (1)  The  difficulty  of  obtaining  a  con- 
stant supply  of  uniformly  intense  light;  (2)  The  uncertainty  of  the  effect  of 
the  position  of  the  screen  with  respect  to  the  incident  beams  and  the  eye  of 
the  observer;  (3)  The  possible  change  in  the  colour  of  the  incident  light  due 
to  the  fluorescence  of  the  substance  of  the  screen.  Professor  Stokes  haa  found 
that  many  substances,  when  illuminated  by  homogeneous  light  of  one  refrangi- 
bility,  become  themselves  luminous,  so  as  to  emit  light  of  lower  refrangibility. 
This  phenomenon  must  be  carefully  attended  to  when  screens  are  used  to  exhibit 
light. 


144  EXPERIMENTS    ON     COLOUK,     AS    PERCEIVED    BY    THE    EYE. 


(3)     Union   of  Coloured   Beams   hy   a   Piism  so   as   to  form   one   Beam. 

The  mode  of  viewing  the  beam  of  light  directly,  without  first  throwing  it 
on  a  screen,  was  not  much  used  by  the  older  experimenters,  but  it  possesses 
the  advantage  of  saving  much  light,  and  admits  of  examining  the  rays  before 
they  have  been  stopped  in  any  way.  In  Newton's  11th  proposition  of  the  2nd 
Book,  an  experiment  is  described,  in  which  a  beam  is  analysed  by  a  prism, 
concentrated  by  a  lens,  and  recombined  by  another  prism,  so  as  to  form  a  beam 
of  white  light  similar  to  the  incident  beam.  By  stopping  the  coloured  rays  at 
the  lens,  any  proposed  combination  may  be  made  to  pass  into  the  emergent 
beam,  where  it  may  be  received  directly  by  the  eye,  or  on  a  screen,  at  pleasure. 

The  experiments  of  Helmholtz  on  the  colours  of  the  spectrum  were  made 
with  the  ordinary  apparatus  for  directly  viewing  the  pure  spectrum,  two  oblique 
slits  crossing  one  another  being  employed  to  admit  the  light  instead  of  one 
vertical  sht.  Two  pure  spectra  were  then  seen  crossing  each  other,  and  so 
exhibiting  at  once  a  large  number  of  combinations.  The  proportions  of  these 
combinations  were  altered  by  varying  the  inclination  of  the  slits  to  the  plane  of 
lefraction,  and  in  this  way  a  number  of  very  remarkable  results  were  obtained, — 
for  which  see  his  Memoir,  before  referred  to. 

In  experiments  of  the  same  kind  made  by  myself  in  August  1852  (inde- 
pendently of  M.  Helmholtz),  I  used  a  combination  of  three  moveable  vertical 
slits  to  admit  the  light,  instead  of  two  cross  shts,  and  observed  the  compound 
ray  through  a  slit  made  in  a  screen  on  which  the  pure  spectrum  is  formed. 
In  this  way  a  considerable  field  of  view  was  filled  with  the  mixed  light,  and 
might  be  compared  with  another  part  of  the  field  illuminated  by  light  proceeding 
from  a  second  system  of  slits,  placed  below  the  first  set.  The  general  character 
of  the  results  agreed  with  those  of  M.  Helmholtz.  The  chief  difficulties  seemed 
to  arise  from  the  defects  of  the  optical  apparatus  of  my  own  eye,  which  ren- 
dered apparent  the  compound  nature  of  the  light,  by  analysing  it  as  a  prism 
or  an  ordinary  lens  would  do,  whenever  the  lights  mixed  differed  much  in 
refrangibility. 


EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE.  145 


(4)     Union   of  two   beams   by   means   of  a   transparent   surface,   which   reflects 
the  first  and  transmits  the  second. 

The  simplest  experiment  of  this  kind  is  described  by  M.  Helmholtz.  He 
places  two  coloured  wafers  on  a  table,  and  then,  taking  a  piece  of  transparent 
glass,  he  places  it  between  them,  so  that  the  reflected  image  of  one  apparently 
coincides  with  the  other  as  seen  through  the  glaas.  The  colours  are  thus  mixed, 
and,  by  varying  the  angle  of  reflection,  the  relative  intensities  of  the  reflected 
and  transmitted  beams  may  be  varied  at  pleasure. 

In  an  instrument  constructed  by  myself  for  photometrical  purposes  two  re- 
flecting plates  were  used.  They  were  placed  in  a  square  tube,  so  as  to  polarize 
the  incident  light,  which  entered  through  holes  in  the  sides  of  the  tubes,  and 
was  reflected  in  the  direction  of  the  axis.  In  this  way  two  beams  oppositely 
polarized  were  mixed,  either  of  which  could  be  coloured  in  any  way  by  coloured 
glasses  placed  over  the  holes  in  the  tube.  By  means  of  a  Nicol's  prism  placed 
at  the  end  of  the  tube,  the  relative  intensities  of  the  two  colours  as  they 
entered  the  eye  could  be  altered  at  pleasure. 

(5)     Union  of  two  coloured  beams  by  means  of  a  doubly -refracting  Prism. 

I  am  not  aware  that  this  method  has  been  tried,  although  the  opposite 
polarization  of  the  emergent  rays  is  favourable  to  the  variation  of  the  experiment. 


(6)     Successive  presentation  of  the  different  Colours  to  the  Retina. 

It  has  long  been  known,  that  light  does  not  produce  its  full  effect  on  the 
eye  at  once,  and  that  the  effect,  when  produced,  remains  visible  for  some  time 
after  the  light  has  ceased  to  act.  In  the  case  of  the  rotating  disc,  the  various 
colours  become  indistinguishable,  and  the  disc  appears  of  a  imiform  tint,  which 
is  in  some  sense  the  resultant  of  the  colours  so  blended.  This  method  of  com- 
bining colours  has  been  used  since  the  time  of  Newton,  to  exhibit  the  results 
of  theory.  The  experiments  of  Professor  J.  D.  Forbes,  which  I  witnessed  in 
1849,  first  encouraged  me  to  think  that  the  laws  of  this  kind  of  mixture  might 
be  discovered  by  special  experiments.  After  repeating  the  well-known  experiment 
in   which   a   series   of  colours   representing   those   of   the   spectrum   are    combined 

VOL.  I.  ^^ 


146  EXPERIMENTS    ON    COLOUB,     AS    PERCEIVED    BY    THE    EYE. 

to  form  gray,  Professor  Forbes  endeavoured  to  form  a  neutral  tint,  by  the 
combination  of  three  colours  only.  For  this  purpose,  he  combined  the  three 
so-called  primary  colours,  red,  blue,  and  yellow,  but  the  resulting  tint  could 
not  be  rendered  neutral  by  any  combination  of  these  colours ;  and  the  reason 
was  found  to  be,  that  blue  and  yellow  do  not  make  green,  but  a  pinkish  tint, 
when  neither  prevails  in  the  combination.  It  was  plain,  that  no  addition  of 
red  to  this,  could  produce  a  neutral  tint. 

This  result  of  mixing  blue  and  yellow  was,  I  beUeve,  not  previously  known. 
It  directly  contradicted  the  received  theory  of  colours,  and  seemed  to  be  at 
variance  with  the  fact,  that  the  same  blue  and  yellow  paint,  when  ground 
together,  do  make  green.  Several  experiments  were  proposed  by  Professor  Forbes, 
in  order  to  eliminate  the  effect  of  motion,  but  he  was  not  then  able  to  under- 
take them.  One  of  these  consisted  in  viewing  alternate  stripes  of  blue  and 
yellow,  with  a  telescope  out  of  focus.  I  have  tried  this,  and  find  the  resultant 
tint  pink  as  before*.  I  also  found  that  the  beams  of  light  coloured  by  trans- 
mission through  blue  and  yellow  glasses  appeared  pink,  when  mixed  on  a  screen, 
while  a  beam  of  light,  after  passing  through  both  glasses,  appeared  green.  By 
the  help  of  the  theory  of  absorption,  given  by  Herschelf,  I  made  out  the 
complete  explanation  of  this  phenomenon.  Those  of  pigments  were,  I  think,  first 
explained  by  Helmholtz  in  the  manner  above  referred  to  J. 

It  may  still  be  asked,  whether  the  effect  of  successive  presentation  to  the 
eye  is  identical  with  that  of  simultaneous  presentation,  for  if  there  is  any  action 
of  the  one  kind  of  light  on  the  other,  it  can  take  place  only  in  the  case  of 
vsimultaneous  presentation.  An  experiment  tending  to  settle  this  point  is  recorded 
by  Newton  (Book  i.  Part  ii.,  Exp.  10).  He  used  a  comb  with  large  teeth  to 
intercept  various  rays  of  the  spectrum.  When  it  was  moved  slowly,  the  various 
colours  could  be  perceived,  but  when  the  speed  was  increased  the  result  was 
perfect  whiteness.  For  another  form  of  this  experiment,  see  Newton's  Sixth 
Letter  to  Oldenburg  (Horsley's  Edition,  Vol.  iv.,  page  335). 

In  order  more  fully  to  satisfy  myself  on  this  subject,  I  took  a  disc  in 
which  were  cut  a  number  of  sUts,  so  as  to  divide  it  into  spokes.  In  a  plane, 
net-rly  passing   through   the   axis  of  this  disc,  I  placed  a  blue  glass,  so  that  one 

*  See  however  Encyc.  Metropolitana,  Art.  "Light,"  section  502.  t  lb.  sect.  516. 

X  I  have  lately  seen  a  passage  in  Moigno's  Cosmos,  stating  that  M.  Plateau,  in  1819,  had  obtained 
jjray  by  whirling  together  gamboge  and  Prussian  blue.  Correspondance  Math,  et  Phys.  de  M.  Quet«let, 
Vol.  v.,  p.  221. 


EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE.  147 

half  of  the  disc  might  be  seen  by  transmitted  light — blue,  and  the  other  by 
reflected  light — white.  In  the  course  of  the  reflected  light  I  placed  a  yellow 
glass,  and  in  this  way  I  had  two  nearly  coincident  images  of  the  slits,  one 
yellow  and  one  blue.  By  turning  the  disc  slowly,  I  observed  that  in  some 
parts  the  yellow  slits  and  the  blue  slits  appeared  to  pass  over  the  field  alter- 
nately, while  in  others  they  appeared  superimposed,  so  as  to  produce  alternately 
their  mixture,  which  was  pale  pink,  and  complete  darkness.  As  long  as  the 
disc  moved  slowly  I  could  perceive  this,  but  when  the  speed  became  great,  the 
whole  field  appeared  uniformly  coloured  pink,  so  that  those  parts  in  which  the 
colours  were  seen  successively  were  indistinguishable  from  those  in  which  they 
were  presented  together  to  the  eye. 

Another  form  in  which  the  experiment  may  be  tried  requires  only  the 
colour-top  above  described.  The  disc  should  be  covered  with  alternate  sectors 
of  any  two  colours,  say  red  and  green,  disposed  alternately  in  four  quadrants. 
By  placing  a  piece  of  glass  above  the  top,  in  the  plane  of  the  axis,  we  make 
the  image  of  one  half  seen  by  reflection  coincide  with  that  of  the  other  seen 
by  transmission.  It  wiU  then  be  seen  that,  in  the  diameters  of  the  top  which 
are  parallel  and  perpendicular  to  the  plane  of  reflection,  the  transmitted  green 
coincides  with  the  reflected  green,  and  the  transmitted  red  with  the  reflected 
red,  so  that  the  result  is  always  either  pure  red  or  pure  green.  But  in  the 
diameters  intermediate  to  these,  the  transmitted  red  coincides  with  the  reflected 
green,  and  vice  versa,  so  that  the  pure  colours  are  never  seen,  but  only  their 
mixtures.  As  long  as  the  top  is  spun  slowly,  these  parts  of  the  disc  will 
appear  more  steady  in  colour  than  those  in  which  the  greatest  alternations 
take  place ;  but  when  the  speed  is  sufficiently  increased,  the  disc  appears  per- 
fectly uniform  in  colour.  From  these  experiments  it  appears,  that  the  apparent 
mixture  of  colours  is  not  due  to  a  mechanical  superposition  of  vibrations,  or 
to  any  mutual  action  of  the  mixed  rays,  but  to  some  cause  residing  in  the 
constitution  of  the  apparatus  of  vision. 


(7)     Presentation  of  the  Colours  to  he  mixed  one  to  each  Eye. 

This  method  is  said  not  to  succeed  with  some  people ;  but  I  have  always 
found  that  the  mixture  of  colours  was  perfect,  although  it  was  difficult  to  con- 
ceive  the    objects   seen   by   the   two   eyes   as   identical.      In   using   the   spectacles, 

19—2 


148  EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE. 

of  which  one  eye  is  green  and  the  other  red,  I  have  found,  when  looking  at 
an  arrangement  of  green  and  red  papers,  that  some  looked  metallic  and  others 
transparent.  This  arises  from  the  very  different  relations  of  brightness  of  the 
two  colours  as  seen  by  each  eye  through  the  spectacles,  which  suggests  the  false 
conclusion,  that  these  differences  are  the  result  of  reflection  from  a  polished 
surface,  or  of  light  transmitted  through  a  clear  one. 


Note  IT. 

Results  of  Experiments  with  Mr  Hay's  Papers  at  Cambridge,  November,  1854. 

The  mean  of  ten  observations  made  by  six  observers  gave 

•449  E+-299  G  +  -252  B=-224  W+776  Bk (l). 

■696  R+-304  G  =  '181  B  +  -327  Y  +  '492  Bk (2). 

These   two  equations  served  to  determine  the  positions  of  white  and  yellow 
in  diagram   No.    2.     The   coeflScient   of  W   is    4*447,    and   that   of  yellow   2'506. 

From    these    data    we    may   deduce   three   other   equations,    either   by   calcu- 
lation,  or  by  measurement  on  the  diagram  (No.    2). 

Eliminating  green  from  the  equations,  we  find 

•565  B  +  -435  Y  =  -307  E.  +  -304  W  +  -389  Bk (3). 

The  mean  of  three  observations  by  three  different  observers  gives 

•573  B-f477  Y  =  ^313  E  +  ^297  W  +  -390Bk. 
Errors  of  calculation    -  '008  B  +  ^008  Y  -  '006  K  +  ^007  W  -  •OOl  Bk. 

The   point   on  the  diagram   to   which   this    equation   corresponds   is   the  intersec- 
tion of  the  lines  BY  and  RW,  and  the  resultant  tint  is  a  pinkish-gray. 

Eliminating  red  from  the  equations,  we  obtain 
Calculation  "533  B-fl50  G-f317  Y  =  ^337  W-f -663  Bk" 

By  10  observations     -537  B-l- '146  G-h  ^317  Y= -337  W-f '663  Bk  ■ (4). 

Errors  -'004      -f- -004  —  —  — 

Eliminating  blue         •660  R-f340  G  =  -218  Y  +  -108  W-f '682  Bkl 

By  5  observations      ^672  R-f '328  G  =  "224  Y+ '094  W-f672  Bk  i (5). 

Errors  -'012      -f012      -•006      -f014       -f008         I 


EXPERIMENTS    ON     COLOUR,     AS    PERCEIVED     BY     THE     EYE.  149 

Note  III. 

On  the  Tlicory  of  Compound  Colours. 

Newton's  theorem  on  the  mixture  of  colours  is  to  be  found  in  his  Opticks, 
Book  I.,  Part  ii.,  Prop.  vi. 

In  a  mixtiu'e  of  primary  colours^  the  quantity  and  quality  of  each  being 
gicen,  to  know  the  colour  of  the  compound. 

He  divides  the  circumference  of  a  circle  into  parts  proportional  to  the  seven 
musical  intervals,  in  accordance  with  his  opinion  of  the  divisions  of  the  spectrum. 
He  then  conceives  the  colours  of  the  spectrum  arranged  round  the  circle,  and  at 
the  centre  of  gravity  of  each  of  the  seven  arcs  he  places  a  little  circle,  the 
area  of  which  represents  the  number  of  rays  of  the  corresponding  colour  which 
enter  into  the  given  mixture.  He  takes  the  centre  of  gravity  of  all  these  circles 
to  represent  the  colour  formed  by  the  mixture.  The  hue  is  determined  by 
drawing  a  line  through  the  centre  of  the  circle  and  this  point  to  the  circum- 
ference. The  position  of  this  line  points  out  the  colour  of  the  spectrum  which 
the  mixture  most  resembles,  and  the  distance  of  the  resultant  tint  from  the 
centre  determines  the  fulness  of  its  colour. 

Newton,  by  this  construction  (for  which  he  gives  no  reasons),  plainly  shews 
that  he  considered  it  possible  to  find  a  place  within  his  circle  for  every  possible 
colour,  and  that  the  entire  nature  of  any  compound  colour  may  be  known  from 
its  place  in  the  circle.  It  will  be  seen  that  the  same  colour  may  be  compounded 
from  the  colours  of  the  spectrum  in  an  infinite  variety  of  ways.  The  apparent 
identity  of  all  these  mixtures,  which  are  optically  different,  as  may  be  shewn  by 
the  prism,  implies  some  law  of  vision  not  explicitly  stated  by  Newton.  This 
law,  if  Newton's  method  be  true,  must  be  that  which  I  have  endeavoured  to 
establish,  namely,  the  threefold  nature  of  sensible  colour. 

With  respect  to  Newton's  construction,  we  now  know  that  the  proportions 
of  the  colours  of  the  spectrum  vary  with  the  nature  of  the  refracting  medium. 
The  only  absolute  index  of  the  kind  of  light  is  the  time  of  its  vibration.  The 
length  of  its  vibration  depends  on  the  medium  in  which  it  is ;  and  if  any  pro- 
portions are  to  be  sought  among  the  wave-lengths  of  the  colours,  they  must 
be   determined   for   those   tissues   of  the   eye   in    which   their   physical  effects  are 


150  EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE. 

supposed  to  terminate.  It  may  be  remarked,  *that  the  apparent  colour  of  the 
spectrum  changes  most  rapidly  at  three  points,  which  lie  respectively  in  the 
yellow,  between  blue  and  green,  and  between  violet  and  blue.  The  wave-lengths 
of  the  corresponding  rays  in  'water  are  in  the  proportions  of  three  geometric 
means  between  1  and  2  very  nearly.  This  result,  however,  is  not  to  be  con- 
sidered established,  unless  confirmed  by  better  observations  than  mine. 

The  only  safe  method  of  completing  Newton's  construction  is  by  an  exami- 
nation of  the  colours  of  the  spectrum  and  their  mixtures,  and  subsequent 
calculation  by  the  method  used  in  the  experiments  with  coloured  papers.  In 
this  way  I  hope  to  determine  the  relative  positions  in  the  colour-diagram  of 
every  ray  of  the  spectrum,  and  its  relative  intensity  in  the  solar  light.  The 
spectrum  will  then  form  a  curve  not  necessarily  circular  or  even  re-entrant,  and 
its  peculiarities  so  ascertained  may  form  the  foundation  of  a  more  complete 
theory  of  the  colour-sensation. 


On  the  relation  of  the  pure  rays  of  the  Spectrum  to  the  three  assumed  Elementary 

Sensations. 

If  we  place  the  three  elementary  colour-sensations  (which  we  may  call,  after 
Young,  red,  green,  and  violet)  at  the  angles  of  a  triangle,  all  colours  which 
the  eye  can  possibly  perceive  (whether  by  the  action  of  light,  or  by  pressure, 
disease,  or  imagination)  must  be  somewhere  within  this  triangle,  those  which  lie 
farthest  from  the  centre  being  the  fullest  and  purest  colours.  Hence  the  colours 
which  lie  at  the  middle  of  the  sides  are  the  purest  of  their  kind  which  the 
eye  can  see,  although  not  so  pure  as  the  elementary  sensations. 

It  is  natural  to  suppose  that  the  pure  red,  green,  and  violet  rays  of  the 
spectrum  produce  the  sensations  which  bear  their  names  in  the  highest  purity. 
But  from  this  supposition  it  would  follow  that  the  yellow,  composed  of  the  red 
and  green  of  the  spectrum,  would  be  the  most  intense  yellow  possible,  while 
it  is  the  result  of  experiment,  that  the  yellow  of  the  spectrum  itself  is  much 
more  full  in  colour.  Hence  the  sensations  produced  by  the  pure  red  and  green 
rays  of  the  spectrum  are  not  the  pure  sensations  of  our  theory.  Newton  has 
remarked,  that  no  two  colours  of  the  spectrum  produce,  when  mixed,  a  colour 
equal  in  fulness  to  the  intermediate  colour.  The  colours  of  the  spectrum  are 
all  more    intense    than    any   compound    ones.      Purple   is   the    only   colour   which 


EXrERIMENTS     ON    COLOUR,     AS    PERCEIVED     BY    TUE     EYE.  151 

must  be  produced  by  combination.  The  experiments  of  Helmholtz  lead  to  the 
same  conclusion ;  and  hence  it  would  appear  that  we  can  find  no  part  of  the 
spectrum  which  produces  a  pure  sensation. 

An  additional,  though  less  satisfactory  evidence  of  this,  is  supplied  by  the 
observation  of  the  colours  of  the  spectrum  when  excessively  bright.  They  then 
appear  to  lose  their  peculiar  colour,  and  to  merge  into  pure  whiteness.  This 
is  probably  due  to  the  want  of  capacity  of  the  organ  to  take  in  so  strong  an 
impression ;  one  sensation  becomes  first  saturated,  and  the  other  two  speedily 
follow  it,  the  final  efiect  being  simple  brightness. 

From  these  facte  I  would  conclude,  that  every  ray  of  the  spectrum  is  capable 
of  producing  all  three  pure  sensations,  though  in  different  degrees.  The  curve, 
therefore,  which  we  have  supposed  to  represent  the  spectrum  will  be  quite  within 
the  triangle  of  colour.  All  natural  or  artificial  colours,  being  compounded  of 
the  colours  of  the  spectrum,  must  lie  within  this  curve,  and,  therefore,  the  colours 
corresponding  to  those  parts  of  the  triangle  beyond  this  curve  must  be  for  ever 
unknown  to  us.  The  determination  of  the  exact  nature  of  the  pure  sensations, 
or  of  their  relation  to  ordinary  colours,  is  therefore  impossible,  unless  we  can 
prevent  them  from  interfering  with  each  other  as  they  do.  It  may  be  possible 
to  experience  sensations  more  pure  than  those  directly  produced  by  the  spec- 
trum, by  first  exhausting  the  sensibility  to  one  colour  by  protracted  gazing,  and 
then  suddenly  turning  to  its  opposite.  But  if,  as  I  suspect,  colour-blindness  be 
due  to  the  absence  of  one  of  these  sensations,  then  the  point  D  in  diagram  (2), 
which  indicates  their  absent  sensation,  indicates  also  our  pure  sensation,  which 
we  may  call  red,  but  which  we  can  never  experience,  because  all  kinds  of 
light  excite  the  other  sensations. 

Newton  has  stated  one  objection  to  his  theory,  as  follows: — "Also,  if  only 
two  of  the  pnmanj  colours,  which  in  tJw  circle  are  opposite  to  one  another,  be 
mixed  in  an  equal  proportion,  the  point  Z"  (the  resultant  tint)  "shall  fall  upon 
the  centre  0 "  (neutral  tint)  ;  "  and  yet  the  colour  compounded  of  these  two  shcdl 
not  he  p>erfectly  white,  hut  some  faint  anonymous  colour.  For  I  could  never  yet,  by 
mixing  only  two  primary  colours,  produce  a  perfect  ivhite"  This  is  confirmed  by 
the  experiments  of  Helmholtz ;  who,  however,  has  succeeded  better  with  some 
pairs  of  colours   than  with  others. 

In.  my  experiments  on  the  spectrum,  I  came  to  the  same  result ;  but  It 
appeared   to   me  that  the  very  peculiar  appearance  of  the  neutral  tints  produced 


152  EXPERIMENTS    ON    COLOUR,     AS    PERCEIVED    BY    THE    EYE. 

was  owing  to  some  opticjal  effect  taking  place  in  the  transparent  part  of  the 
eye  on  the  mixture  of  two  rays  of  very  different  refrangibility.  Most  eyes  are 
by  no  means  achromatic,  so  that  the  images  of  objects  illuminated  with  mixed 
light  of  this  kind  appear  divided  into  two  different  colours;  and  even  when 
there  is  no  distinct  object,  the  mixtures  become  in  some  degree  analysed,  so  as 
to  present  a  very  strange,  and  certainly  "anonymous"  appearance. 


Additional  Note  on  the  more  recent  experiments  of  M.  Helmholtz*. 

In  his  former  memoir  on  the  Theory  of  Compound  Colours  f,  M.  Helmholtz 
arrived  at  the  conclusion  that  only  one  pair  of  homogeneous  colours,  orange- 
yellow  and  indigo-blue,  were  strictly  complementary.  This  result  was  shewn  by 
Professor  Grassmann|  to  be  at  variance  with  Newton's  theory  of  compound 
colours ;  and  although  the  reasoning  was  founded  on  intuitive  rather  than 
experimental  truths,  it  pointed  out  the  tests  by  which  Newton's  theory  must 
be  verified  or  overthrown.  In  applying  these  tests,  M.  Helmholtz  made  use  of 
an  apparatus  similar  to  that  described  by  M.  Foucault§,  by  which  a  screen  of 
white  paper  is  illuminated  by  the  mixed  light.  The  field  of  mixed  colour  is 
much  larger  than  in  M.  Helmholtz's  former  experiments,  and  the  facility  of 
forming  combinations  is  much  increased.  In  this  memoir  the  mathematical  theory 
of  Newton's  circle,  and  of  the  curve  formed  by  the  spectrum,  with  its  possible 
transformations,  is  completely  stated,  and  the  form  of  this  curve  is  in  some 
degree  indicated,  as  far  as  the  determination  of  the  colours  which  he  on  oppo- 
site sides  of  white,  and  of  those  which  He  opposite  the  part  of  the  curve  which 
is  wanting.  The  colours  between  red  and  yellow-green  are  complementary  to 
colours  between  blue-green  and  violet,  and  those  between  yellow-green  and  blue- 
green  have  no  homogeneous  complementaries,  but  must  be  neutrahzed  by  various 
hues  of  purple,  i.e.,  mixtures  of  red  and  violet.  The  names  of  the  complementary 
colours,  with  their  wave-lengths  in  air,  as  deduced  from  Fraunhofer's  measure- 
ments, are  given  in  the  following  table  : — 

•  PoggendorflF's  Annalen,  BA  xciv.      (I  am  indebted  for  the  perusal  of  this  Memoir  to  Professor 
Stokes.) 

+  lb.  Bd.  Lxxxvii.     Annals  of  Philosophy,  1852,  Part  ii. 

t  Ih.  Bd.  Lxxxix.     Ann.  Phil.,  1854,  April. 

§  lb.  Bd.  LXixvm.     Moigno,  Cosmos,  1853,  Tom.  ii,,  p.  232. 


EXPERIMENTS  ON  COLOUR,  AS  PERCEIVED  BY  THE  EYE. 


153 


Colour 

Wave-length 

Complementary 
Colour 

Wave-length 

Ratio  of 
wave-lengths 

Red    ...     . 

Orange    .     .     . 
Gold-yellow 
Gold  veUow      . 
Yellow    .     .     . 
Yellow    .     .     . 
Green-yellow     . 

2425 
2244 
2162 
2120 
2095 
2085 
2082 

Green-blue  . 
Blue  .     .     . 
Blue  .     .     . 
Blue  .     .     . 
Indigo-blue 
Indigo-blue 
Violet     .     . 

1818 
1809 
1793 
1781 
1716 
1706 
1600- 

1-334 
1-240 
1-206 
1-190 
1-221 
1-222 
1-301 

(The  wave-lengtha  are  expressed  in  millionths  of  a  Paris  inch.) 

(In  order  to  reduce  these  wave-lengths  to  their  actual  lengths  in  the  eye, 
each  must  be  divided  by  the  index  of  refraction  for  that  kind  of  light  in  the 
medium  in  which  the  physical  etfect  of  the  vibrations  is  supposed  to  take  place.) 

Although  these  experiments  are  not  in  themselves  sufficient  to  give  the  com- 
plete theory  of  the  curve  of  homogeneous  colours,  they  determine  the  most 
important  element  of  that  theory  in  a  way  which  seems  very  accurate,  and  I 
cannot  doubt  that  when  a  philosopher  who  has  so  fully  pointed  out  the  im- 
portance of  general  theories  in  physics  turns  his  attention  to  the  theory  of 
sensation,  he  will  at  least  establish  the  principle  that  the  laws  of  sensation  can 
be  successfully  investigated  only  after  the  corresponding  physical  laws  have  been 
ascertained,  and  that  the  connection  of  these  two  kinds  of  laws  can  be  appre- 
hended only  when  the  distinction  between  them  is  fully  recognised. 


Note  IV. 


Description  of  the  Figures.     Plate  I. 

No.  1.  is  the  colour-diagi-am  already  referred  to,  representing,  cm  Newton's  principle,  the  relations  of 
diflferent  coloured  papers  to  the  three  standard  colours— vermilion,  emerald-green,  and  ultra- 
marine. The  initials  denoting  the  colours  are  explained  in  the  list  at  page  276,  and  the 
numbers  belonging  to  them  are  their  coefficients  of  intensity,  the  use  of  which  has  been 
explained.  The  initials  H.R.,  H.B.,  and  H.G.,  represent  the  red,  blue  and  green  papers 
of  Mr  Hay,  and  serve  to  connect  this  diagram  -vith  No.  (2),  which  takes  these  colours  for 
its  standards. 
VOL.  I.  20 


154  EXPEBIMENTS  ON  COLOUK,  AS  PERCEIVED  BY  THE  EYE. 

No.  2.  represents  the  relations  of  Mr  Hay's  red,  blue,  green,  white,  and  yellow  papers,  as  deter- 
mined by  a  large  number  of  experiments  at  Cambridge. — (See  Note  II.).  The  use  of  the 
point   D,    in   calculating   the   results   of  colour-blindness,    is   explained   in   the   Paper. 

Fig,  3.  represents   a   disc   of   the   larger  size,    with   its   slit. 

Fig.  4.  shows   the   mode   of  combining   two   discs   of   the   smaller   size. 

Fi«^.  5.  shows   the   combination   of  discs,   as  placed   on    the    top,    in    the    first    experiment    described 

in   the   Paper. 
Fig.  6.  represents   the   method   of  spinning  the   top,   when   speed   is   required. 
The   last   four   figures   are   half  the   actual   size. 

Colour-tops  of  the  kind  used  in  these  experiments,  with  paper  discs  of  the  colours  whose  relations 
are   represented   in   No.    1,   are   to   be   had   of  Mr  J.    M.    Bryson,    Optician,    Edinburgh. 


VOL.  I.  PLATE  L 


Mue,-,^^^ 


^ 


^ 


/ 


C04 


% 


HB06*     i 


UCO-4 


^GO-2     / 


^' 


^" 


^^^. 


(1^?1 


tUTyJO 


li^C. 


^6reen/lo 


VOL,  L  PLATE  L 


riG^.4 


Pia.3 


FIG.6 


FIG.  6 


[From  the  Transactions  of  the  Cambridge  Philosophical  Society,  VoL  x.  Part  i.] 

VIII.      On    Faraday's  Lines  of  Force. 
[Read  Dec.  10,  1855,  and  Feb.  11,  1856.] 

The  present  state  of  electrical  science  seems  peculiarl^^  unfavourable  to  specu- 
lation. The  laws  of  the  distribution  of  electricity  on  the  surface  of  conductors 
have  been  analytically  deduced  from  experiment;  some  parts  of  the  mathematical 
theory  of  magnetism  are  established,  while  in  other  parts  the  experimental  data 
are  wanting ;  the  theory  of  the  conduction  of  galvanism  and  that  of  the  mutual 
attraction  of  conductors  have  been  reduced  to  mathematical  formulae,  but  have 
not  fallen  into  relation  with  the  other  parts  of  the  science.  No  electrical  theory 
can  now  be  put  forth,  unless  it  shews  the  connexion  not  only  between  electricity 
at  rest  and  current  electricity,  but  between  the  attractions  and  inductive  effects 
of  electricity  in  both  states.  Such  a  theory  must  accurately  satisfy  those  laws, 
the  mathematical  form  of  which  is  known,  and  must  afford  the  means  of  calcu- 
lating the  effects  in  the  limiting  cases  where  the  known  formulae  are  inapplicable. 
In  order  therefore  to  appreciate  the  requirements  of  the  science,  the  student 
must  make  himself  familiar  with  a  considerable  body  of  most  intricate  mathe- 
matics, the  merfi  retention  of  which  in  the  memory  materially  interferes  with 
further  progress.  The  first  process  therefore  in  the  effectual  study  of  the  science^, 
must  be  one  of  simplification  and  reduction  of  the  results  of  previous  investiga- 
tion to  a  form  in  which  the  mind  can  grasp  them.  The  results  of  this  simplifi- 
cation may  take  the  form  of  a  purely  mathematical  formula  or  of  a  physical 
hypothesis.  In  the  first  case  we  entirely  lose  sight  of  the  phenomena  to  be 
explained ;  and  though  we  may  trace  out  the  consequences  of  given  laws,  we 
can  never  obtain  more  extended  views  of  the  connexions  of  the  subject^  If, 
on  the  other  luiml,  we  adopt  a  physical  hypothesis,  we  see  the  phenomena  only 
throucrh    a    medium,    and    are    liable    to    that    blindness    to    facts    and    rashness    m 

20—2 


156  ON    FARADAY  S    LINES    OF    FdRCE. 

assumption  wKich  a  partial  explanation  encourages.  "We  must  therefore  discover 
some  method  of  investigation  which  allows  the  mind  at  every  step  to  lay  hold 
of  a  clear  physical  conception,  without  being  committed  to  any  theory  founded 
on  the  physical  science  from  which  that  conception  is  borrowed,  so  that  it  is 
neither  drawn  aside  from  the  subject  in  pursuit  of  analytical  subtleties,  nor  carried 
beyond  the  truth  by  a  favourite  hypothesis. 

In  order  to  obtain  physical  ideas  without  adopting  a  physical  theory  we  must 
make  ourselves  familiar  with  the  existence  of  physical  analogies.  By  a  physical 
analogy  I  mean  that  partial  similarity  between  the  laws  of  one  science  and  those 
of  another  which  makes  each  of  them  illustrate  the  other.  Thus  all  the  mathe- 
matical sciences  are  founded  on  relations  between  physical  laws  and  laws  of 
numbers,  so  that  the  aim  of  exact  science  is  to  reduce  the  problems  of  nature 
to  the  determination  of  quantities  by  operations  with  numbers.  Passing  from 
the  most  universal  of  all  analogies  to  a  very  partial  one,  we  find  the  same 
resemblance  in  mathematical  form  between  two  different  phenomena  giving  rise 
to  a  physical  theory  of  light. 

The  changes  of  direction  which  light  undergoes  in  passing  from  one  medium 
to  another,  are  identical  with  the  deviations  of  the  path  of  a  particle  in  moving 
through  a  narrow  space  in  which  intense  forces  act.  This  analogy,  which  extends 
only  to  the  direction,  and  not  to  the  velocity  of  motion,  was  long  believed  to 
he  the  true  explanation  of  the  refraction  of  Ught ;  and  we  still  find  it  useful 
in  the  solution  of  certain  problems,  in  which  we  employ  it  without  danger,  as 
an  artificial  method.  The  other  analogy,  between  light  and  the  vibrations  of  an 
elastic  medium,  extends  much  farther,  but,  though  its  importance  and  fruitfulness 
cannot  be  over-estimated,  we  must  recollect  that  it  is  founded  only  on  a  resem- 
blance in  form  between  the  laws  of  light  and  those  of  vibrations.  By  stripping 
it  of  its  physical  dress  and  reducing  it  to  a  theory  of  "  transverse  alternations," 
we  might  obtain  a  system  of  truth  strictly  founded  on  observation,  but  probably 
deficient  both  in  the  vividness  of  its  conceptions  and  the  fertility  of  its  method. 
I  have  said  thus  much  on  the  disputed  questions  of  Optics,  as  a  preparation 
for  the  discussion  of  the  almost  universally  admitted  theory  of  attraction  at  a 
distance. 

We  have  all  acquired  the  mathematical  conception  of  these  attractions.  We 
can  reason  about  them  and  determine  their  appropriate  forms  or  formulae.  These 
formulae  have  a  distinct  mathematical  significance,  and  their  results  are  found 
to  be  in  accordance   with   natural   phenomena.      There   is   no   formula   in   applied 


ON     FARADAY'3    lines    OF    FORCE.  157 

mathematics  more  consistent  with  nature  than  the  formula  of  attractions,  and  no 
theory  better  estabUshed  in  the  minds  of  men  than  that  of  the  action  of  bodies 
on  one  another  at  a  distance.  The  laws  of  the  conduction  of  heat  in  uniform 
media  appear  at  first  sight  among  the  most  different  in  their  physical  relations 
from  those  relating  to  attractions.  The  quantities  which  enter  into  them  are 
teviperature,  flow  of  heat,  conductivity.  The  word  force  is  foreign  to  the  subject. 
Yet  we  find  that  the  mathematical  laws  of  the  uniform  motion  of  heat  in 
homogeneous  media  are  identical  in  form  with  those  of  attractions  varying  in- 
versely as  the  square  of  the  distance.  We  have  only  to  substitute  source  of 
heat  for  centre  of  attrax^tion,  flow  of  heat  for  accelerating  effect  of  attraction  at 
any  point,  and  temperature  for  potential,  and  the  solution  of  a  problem  in 
attractions   is   transformed  into  that  of  a  problem  in  heat. 

This  analogy  between  the  formulae  of  heat  and  attraction  was,  I  believe, 
first  pointed  out  by  Professor  William  Thomson  in  the  Camh.  Math.  Journal, 
Vol.   III. 

Now  the  conduction  of  heat  is  supposed  to  proceed  by  an  action  between 
contiguous  parts  of  a  medium,  while  the  force  of  attraction  is  a  relation  be- 
tween distant  bodies,  and  yet,  if  we  knew  nothing  more  than  is  expressed  in 
the  mathematical  formulae,  there  would  be  nothing  to  distinguish  between  the 
one  set  of  phenomena  and  the  other. 

It  is  true,  that  if  we  introduce  other  considerations  and  observe  additional 
facts,  the  two  subjects  will  assume  very  difierent  aspects,  but  the  mathematical 
resemblance  of  some  of  their  laws  will  remain,  and  may  still  be  made  useful 
in  exciting   appropriate  mathematical  ideas. 

It  is  by  the  use  of  analogies  of  this  kind  that  I  have  attempted  to  bring 
before  the  mind,  in  a  convenient  and  manageable  form,  those  mathematical  ideas 
which  are  necessary  to  the  study  of  the  phenomena  of  electricity.  The  methods 
are  generally  those  suggested  by  the  processes  of  reasoning  which  are  found  in 
the  researches  of  Faraday"*',  and  which,  though  they  have  been  interpreted 
mathematically  by  Prof.  Thomson  and  others,  are  very  generally  supposed  to  be 
of  an  indefinite  and  unmathematical  character,  when  compared  with  those  em- 
ployed by  the  professed  mathematicians.  By  the  method  which  I  adopt,  I  hope 
to  render  it  evident  that  I  am  not  attempting  to  estabhsh  any  physical  theory 
of  a  science  in  which  I  have  hardly  made  a  single  experiment,  and  that  the 
limit   of  my    design   is  to   shew   how,   by   a   strict   application   of   the   ideas   and 

*  See  especially  Series  xxxviii.  of  the  Experimental  Researcltes,  and  Phil.   Mag.   1852. 


158  ON  Faraday's  lines  of  force. 

methods  of  Faraday,  the  connexion  of  the  very  different  orders  of  phenomena 
which  he  has  discovered  may  be  clearly  placed  before  the  mathematical  mind. 
I  shall  therefore  avoid  as  much  as  I  can  the  introduction  of  anything  which 
does  not  serve  as  a  direct  illustration  of  Faraday's  methods,  or  of  the  mathe- 
matical deductions  which  may  be  made  from  them.  In  treating  the  simpler 
parts  of  the  subject  I  shall  use  Faraday's  mathematical  methods  as  well  as 
his  ideas.  When  the  complexity  of  the  subject  requires  it,  I  shall  use  analytical 
notation,  still  confining  myself  to  the  development  of  ideas  originated  by  the 
same   philosopher. 

I  have  in  the  first  place  to  explain  and  illustrate  the  idea  of  "lines  of 
force." 

When  a  body  is  electrified  in  any  manner,  a  small  body  charged  with  posi- 
tive electricity,  and  placed  in  any  given  position,  will  experience  a  force  urging 
it  in  a  certain  direction.  If  the  small  body  be  now  negatively  electrified,  it  will 
be  urged  by  an  equal  force  in  a  direction  exactly  opposite. 

The  same  relations  hold  between  a  magnetic  body  and  the  north  or  south 
poles  of  a  small  magnet.  If  the  north  pole  is  urged  in  one  direction,  the  south 
pole  is  urged  in  the  opposite  direction. 

In  this  way  we  might  find  a  line  passing  through  any  point  of  space,  such 
that  it  represents  the  direction  of  the  force  acting  on  a  positively  electrified 
particle,  or  on  an  elementary  north  pole,  and  the  reverse  direction  of  the  force 
on  a  negatively  electrified  particle  or  an  elementary  south  pole.  Since  at  every 
point  of  space  such  a  direction  may  be  found,  if  we  commence  at  any  point 
and  draw  a  line  so  that,  as  we  go  along  it,  its  direction  at  any  point  shall 
always  coincide  with  that  of  the  resultant  force  at  that  point,  this  curve  wiU 
indicate  the  direction  of  that  force  for  every  point  through  which  it  passes,  and 
might  be  called  on  that  account  a  line  of  force.  We  might  in  the  same  way 
draw  other  lines  of  force,  till  we  had  filled  all  space  with  curves  indicating  by 
their  direction  that  of  the  force  at  any  assigned  point. 

We  should  thus  obtain  a  geometrical  model  of  the  physical  phenomena, 
which  would  tell  us  the  direction  of  the  force,  but  we  should  stiU  require  some 
method  of  indicating  the  intensity  of  the  force  at  any  point.  If  we  consider 
these  curves  not  as  mere  lines,  but  as  fine  tubes  of  variable  section  carrying 
an  incompressible  fluid,  then,  since  the  velocity  of  the  fluid  is  inversely  as  the 
section  of  the  tube,  we  may  make  the  velocity  vary  according  to  any  given  law, 
by  regulating  the   section  of  the  tube,  and  in  this  way  we  might  represent  the 


ON    FARADAY  S    LINES    OF    FORCE. 


159 


intensity  of  the  force  as  well  as  its  direction  by  the  motion  of  the  fluid  in 
these  tubes.  This  method  of  representing  the  intensity  of  a  force  by  the  velocity 
of  an  imaginary  fluid  in  a  tube  is  applicable  to  any  conceivable  system  of  forces, 
but  it  is  capable  of  great  simplification  in  the  case  in  which  the  forces  are  such 
as  can  be  explained  by  the  hypothesis  of  attractions  varying  inversely  as  the 
square  of  the  distance,  such  as  those  observed  in  electrical  and  magnetic  pheno- 
mena. In  the  case  of  a  perfectly  arbitrary  system  of  forces,  there  will  generally 
be  interstices  between  the  tubes  ;  but  in  the  case  of  electric  and  magnetic  forces 
it  is  possible  to  arrange  the  tubes  so  as  to  leave  no  interstices.  The  tubes  will 
then  be  mere  surfaces,  directing  the  motion  of  a  fluid  filling  up  the  whole  space. 
It  has  been  usual  to  commence  the  investigation  of  the  laws  of  these  forces  by 
at  once  assuming  that  the  phenomena  are  due  to  attractive  or  repulsive  forces 
acting  between  certain  points.  We  may  however  obtain  a  different  view  of  the 
subject,  and  one  more  suited  to  our  more  difficult  inquiries,  by  adopting  for  the 
definition  of  the  forces  of  which  we  treat,  that  they  may  be  represented  in 
magnitude  and  direction  by  the  uniform  motion  of  an  incompressible  fluid. 

I  propose,  then,  first  to  describe  a  method  by  which  the  motion  of  such  a 
fluid  can  be  clearly  conceived;  secondly  to  trace  the  consequences  of  assuming 
certain  conditions  of  motion,  and  to  point  out  the  application  of  the  method  to 
some  of  the  less  complicated  phenomena  of  electricity,  magnetism,  and  galvanism ; 
and  lastly  to  shew  how  by  an  extension  of  these  methods,  and  the  introduction 
of  another  idea  due  to  Faraday,  the  laws  of  the  attractions  and  inductive  actions 
of  magnets  and  currents  may  be  clearly  conceived,  without  making  any  assump- 
tions as  to  the  physical  nature  of  electricity,  or  adding  anything  to  that  which 
has  been  already  proved  by  experiment. 

By  referring  everything  to  the  purely  geometrical  idea  of  the  motion  of  an 
imaginary  fluid,  I  hope  to  attain  generahty  and  precision,  and  to  avoid  the 
dangers  arising  from  a  premature  theory  professing  to  explain  the  cause  of  the 
phenomena.  If  the  results  of  mere  speculation  which  I  have  collected  are  found 
to  be  of  any  use  to  experimental  philosophers,  in  arranging  and  interpreting 
their  results,  they  will  have  served  their  purpose,  and  a  mature  theory,  in  which 
physical  facts  will  be  physically  explained,  will  be  formed  by  those  who  by 
interrogating  Nature  herself  can  obtain  the  only  true  solution  of  the  questions 
which  the  mathematical  theory  suggests. 


160  ON     FARADAY  S    LINES    OF    FORCE. 


I.     Theoi-y  of  the  Motion  of  an  incompressible  Fluid. 

(1)  The  substance  here  treated  of  must  not  be  assumed  to  possess  any  of 
the  properties  of  ordinary  fluids  except  those  of  freedom  of  motion  and  resistance 
to  compression.  It  is  not  even  a  hypothetical  fluid  which  is  introduced  to 
explain  actual  phenomena.  It  is  merely  a  collection  of  imaginary  properties 
which  may  be  employed  for  establishing  certain  theorems  in  pure  mathematics  in 
a  way  more  intelligible  to  many  minds  and  more  applicable  to  physical  problems 
than  that  in  which  algebraic  symbols  alone  are  used.  The  use  of  the  word 
"Fluid"  will  not  lead  us  into  error,  if  we  remember  that  it  denotes  a  purely 
imaginary  substance  with  the  following  property  : 

The  poHion  of  fluid  which  at  any  iTistant  occupied  a  given  volume,  will  at 
any  succeeding  instant  occupy  an  equal  volume. 

This  law  expresses  the  incompressibility  of  the  fluid,  and  furnishes  us  with 
a  convenient  measure  of  its  quantity,  namely  its  volume.  The  unit  of  quantity 
of  the  fluid  will  therefore  be  the  unit  of  volume. 

(2)  The  direction  of  motion  of  the  fluid  will  in  general  be  dlflerent  at 
different  points  of  the  space  which  it  occupies,  but  since  the  direction  is  deter- 
minate for  every  such  point,  we  may  conceive  a  line  to  begin  at  any  point  and 
to  be  continued  so  that  every  element  of  the  line  indicates  by  its  direction  the 
direction  of  motion  at  that  point  of  space.  Lines  drawn  in  such  a  manner  that 
their  direction  always  indicates  the  direction  of  fluid  motion  are  called  lines  of 
fluid  motion. 

If  the  motion  of  the  fluid  be  what  is  called  steady  motion,  that  is,  if  the 
direction  and  velocity  of  the  motion  at  any  fixed  point  be  independent  of  the 
time,  these  curves  will  represent  the  paths  of  individual  particles  of  the  fluid, 
but  if  the  motion  be  variable  this  will  not  generally  be  the  case.  The  cases 
of  motion  which  will  come  under  our  notice  will  be  those  of  steady  motion. 

(3)  If  upon  any  surface  which  cuts  the  lines  of  fluid  motion  we  draw  a 
closed  curve,  and  if  from  every  point  of  this  curve  we  draw  a  line  of  motion, 
these  lines  of  motion  will  generate  a  tubular  surface  which  we  may  call  a  tube 
of  fluid  motion.     Since  this  surface  is  generated  by  lines  in  the  direction  of  fluid 


ON   Faraday's  lines  of  force.  161 

motion    no   part    of  the    fluid    can    flow   across    it,    so   that   this    imaginary  surface 
is  as  impermeable  to  the  fluid  as  a  real  tube. 

(4)  The  quantity  of  fluid  which  in  unit  of  time  crosses  any  fixed  section 
of  the  tube  is  the  same  at  whatever  part  of  the  tube  the  section  be  taken. 
For  the  fluid  is  incompressible,  and  no  part  runs  through  the  sides  of  the  tube, 
therefore  the  quantity  which  escapes  from  the  second  section  is  equal  to  that 
which  enters  through  the  first. 

If  the  tube  be  such  that  unit  of  volume  passes  through  any  section  in 
unit  of  time  it  is  called  a  unit  tube  of  fluid  motion. 

(5)  In  what  follows,  various  units  will  be  referred  to,  and  a  finite  number 
of  lines  or  surfaces  will  be  drawn,  representing  in  terms  of  those  units  the 
motion  of  the  fluid.  Now  in  order  to  define  the  motion  in  every  part  of  the 
fluid,  an  infinite  number  of  lines  would  have  to  be  drawn  at  indefinitely  small 
intervals ;  but  since  the  description  of  such  a  system  of  lines  would  involve 
continual  reference  to  the  theory  of  limits,  it  has  been  thought  better  to  suppose 
the  lines  drawn  at  intervals  depending  on  the  assumed  unit,  and  afterwards  to 
assume  the  unit  as  small  as  we  please  by  taking  a  small  submultiple  of  the 
standard  unit. 

(6)  To  define  the  motion  of  the  whole  fluid  by  means  of  a  system  of  unit 
tubes. 

Take  any  fixed  surface  which  cuts  all  the  lines  of  fluid  motion,  and  draw 
upon  it  any  system  of  curves  not  intersecting  one  another.  On  the  same  surface 
draw  a  second  system  of  curves  intersecting  the  first  system,  and  so  arranged 
that  the  quantity  of  fluid  which  crosses  the  surface  within  each  of  the  quadri- 
laterals formed  by  the  intersection  of  the  two  systems  of  curves  shall  be  unity 
in  unit  of  time.  From  every  point  in  a  curve  of  the  first  system  let  a  line 
of  fluid  motion  be  drawn.  These  lines  will  form  a  surface  through  which  no 
fluid  passes.  Similar  impermeable  surfaces  may  be  drawn  for  all  the  curves  of 
the  first  system.  The  curves  of  the  second  system  will  give  rise  to  a  second 
system  of  impermeable  surfaces,  which,  by  their  intersection  with  the  first  system, 
will  form  quadrilateral  tubes,  which  will  be  tubes  of  fluid  motion.  Since  each 
quadrilateral  of  the  cutting  surface  transmits  unity  of  fluid  in  unity  of  time, 
every  tube  in  the  system  will  transmit  unity  of  fluid  through  any  of  its  sections 
in  unit  of  time.      The  motion  of  the  fluid  at  every  part  of  the  space  it  occupies 

VOL,  I.  21 


162  ON    FARADAY  S    LINES    OF    FORCE. 

is  determined  by  this  system  of  unit  tubes ;  for  the  direction  of  motion  is  that 
of  the  tube  through  the  point  in  question,  and  the  velocity  is  the  reciprocal 
of  the  area  of  the  section  of  the  unit  tube  at.  that  point. 

(7)  We  have  now  obtained  a  geometrical  construction  which  completely 
defines  the  motion  of  the  fluid  by  dividing  the  space  it  occupies  into  a  system 
of  unit  tubes.  We  have  next  to  shew  how  by  means  of  these  tubes  we  may 
ascertain  various  points  relating  to  the  motion  of  the  fluid. 

A  unit  tube  may  either  return  into  itself,  or  may  begin  and  end  at  differ- 
ent points,  and  these  may  be  either  in  the  boundary  of  the  space  in  which  we 
investigate  the  motion,  or  within  that  space.  In  the  first  case  there  is  a  con- 
tinual circulation  of  fluid  in  the  tube,  in  the  second  the  fluid  enters  at  one  end 
and  flows  out  at  the  other.  If  the  extremities  of  the  tube  are  in  the  bound- 
ing surface,  the  fluid  may  be  supposed  to  be  continually  supplied  from  without 
from  an  unknown  source,  and  to  flow  out  at  the  other  into  an  unknown  reser- 
voir;  but  if  the  origin  of  the  tube  or  its  termination  be  within  the  space  under 
consideration,  then  we  must  conceive  the  fluid  to  be  supplied  by  a  source  within 
that  space,  capable  of  creating  and  emitting  unity  of  fluid  in  unity  of  time,  and 
to  be  afterwards  swallowed  up  by  a  sink  capable  of  receiving  and  destroying 
the  same  amount  continually. 

There  is  nothing  self-contradictory  in  the  conception  of  these  sources  where 
the  fluid  is  created,  and  sinks  where  it  is  annihilated.  The  properties  of  the 
fluid  are  at  our  disposal,  we  have  made  it  incompressible,  and  now  we  suppose 
it  produced  from  nothing  at  certain  points  and  reduced  to  nothing  at  others. 
The  places  of  production  will  be  called  sources,  and  their  numerical  value  will  be 
the  number  of  units  of  fluid  which  they  produce  in  unit  of  time.  The  places 
of  reduction  will,  for  want  of  a  better  name,  be  called  sinks,  and  will  be  esti- 
mated by  the  number  of  units  of  fluid  absorbed  in  unit  of  time.  Both  places 
win  sometimes  be  called  sources,  a  source  being  understood  to  be  a  sink  when 
its  sign  is  negative. 

(8)  It  is  evident  that  the  amount  of  fluid  which  passes  any  fixed  surface 
is  measured  by  the  number  of  unit  tubes  which  cut  it,  and  the  direction  in 
which  the  fluid  passes  is  determined  by  that  of  its  motion  in  the  tubes.  If 
the  surface  be  a  closed  one,  then  any  tube  whose  terminations  lie  on  the  same 
side  of  the  surface  must  cross  the  surface  as  many  times  in  the  one  direction 
as  in  the   other,  and  therefore   must   cany  as  much  fluid  out   of  the  surface  as 


ON  Faraday's  lines  of  force.  163 

it  carries  in.  A  tube  which  begins  within  the  surface  and  ends  without  it 
will  carry  out  unity  of  fluid;  and  one  which  enters  the  surface  and  terminates 
within  it  will  carry  in  the  same  quantity.  In  order  therefore  to  estimate  the 
amount  of  fluid  which  flows  out  of  the  closed  surface,  we  must  subtract  the 
number  of  tubes  which  end  within  the  surface  from  the  number  of  tubes  which 
begin  there.     If  the  result  is  negative  the  fluid  will  on  the  whole  flow  inwards. 

If  we  call  the  beginning  of  a  unit  tube  a  unit  source,  and  its  termination 
a  unit  sink,  then  the  quantity  of  fluid  produced  within  the  surface  is  estimated 
by  the  number  of  unit  sources  minus  the  number  of  unit  sinks,  and  this  must 
flow  out  of  the  surface  on  account  of  the  incompressibility  of  the  fluid. 

In  speaking  of  these  imit  tubes,  sources  and  sinks,  we  must  remember  what 
was  stated  in  (5)  as  to  the  magnitude  of  the  unit,  and  how  by  diminishing 
their  size  and  increasing  their  number  we  may  distribute  them  according  to  any 
law   however  complicated. 

(9)  If  we  know  the  direction  and  velocity  of  the  fluid  at  any  point  in 
two  diSerent  cases,  and  if  we  conceive  a  third  case  in  which  the  direction  and 
velocity  of  the  fluid  at  any  point  is  the  resultant  of  the  velocities  in  the  two 
former  cases  at  corresponding  points,  then  the  amount  of  fluid  which  passes  a 
given  fixed  surface  in  the  third  case  will  be  the  algebraic  sum  of  the  quantities 
which  pass  the  same  surface  in  the  two  former  cases.  For  the  rate  at  which 
the  fluid  crosses  any  surface  is  the  resolved  part  of  the  velocity  normal  to  the 
surface,  and  the  resolved  part  of  the  resultant  is  equal  to  the  sum  of  the 
resolved  parts  of  the  components. 

Hence  the  number  of  unit  tubes  which  cross  the  surface  outwards  in  the 
third  case  must  be  the  algebraical  sum  of  the  numbers  which  cross  it  in  the 
two  former  cases,  and  the  number  of  sources  within  any  closed  surface  will  be 
the  sum  of  the  numbers  in  the  two  former  cases.  Since  the  closed  surface  may 
be  taken  as  small  as  we  please,  it  is  evident  that  the  distribution  of  sources 
and  sinks  in  the  third  case  arises  from  the  simple  superposition  of  the  distri- 
butions in  the  two  former  cases. 


n.     TTieory  of  the  uniform  motion  of  an  imponderable  incompressible  fluid 
through  a  resisting  medium. 

(10)    The  fluid  is  here  supposed  to  have  no  inertia,  and  its  motion  is  opposed 
by  the  action  of  a  force  which  we  may  conceive  to  be  due  to  the  resistance  of  a 

21—2 


164  ON    FARADAY  S    LINES    OF    FORCK 

medium  through  which  the  fluid  is  supposed  to  flow.  This  resistance  depends  on 
the  nature  of  the  medium,  and  will  in  general  depend  on  the  direction  in  which 
the  fluid  moves,  as  well  as  on  its  velocity.  For  the  present  we  may  restrict 
ourselves  to  the  case  of  a  uniform  medium,  whose  resistance  is  the  same  in  all 
directions.     The  law  which  we  assume  is  as  follows. 

Any  portion  of  the  fluid  moving  through  the  resisting  medium  is  directly 
opposed  by  a  retarding  force  proportional  to  its  velocity. 

If  the  velocity  be  represented  by  i',  then  the  resistance  will  be  a  force  equal 
to  kv  acting  on  unit  of  volume  of  the  fluid  in  a  direction  contrary  to  that  of 
motion.  In  order,  therefore,  that  the  velocity  may  be  kept  up,  there  must  be  a 
greater  pressure  behind  any  portion  of  the  fluid  than  there  is  in  front  of  it,  so 
that  the  difference  of  pressures  may  neutrahse  the  effect  of  the  resistance.  Con- 
ceive a  cubical  unit  of  fluid  (which  we  may  make  as  small  as  we  please,  by  (5)), 
and  let  it  move  in  a  direction  perpendicular  to  two  of  its  faces.  Then  the  resist- 
ance will  be  kv,  and  therefore  the  difference  of  pressures  on  the  first  and  second 
faces  is  kv,  so  that  the  pressure  diminishes  in  the  direction  of  motion  at  the  rate 
of  kv  for  every  unit  of  length  measured  along  the  line  of  motion ;  so  that  if  w6 
measure  a  length  equal  to  h  units,  the  difference  of  pressure  at  its  extremities 
will  be  kvh. 

(11)  Since  the  pressure  is  supposed  to  vary  continuously  in  the  fluid,  all 
the  points  at  which  the  pressure  is  equal  to  a  given  pressure  p  will  lie  on  a 
certain  surface  which  we  may  call  the  surface  (p)  of  equal  pressure.  If  a  series 
of  these  surfaces  be  constructed  in  the  fluid  corresponding  to  the  pressures  0,  1, 
2,  3  &c.,  then  the  number  of  the  surface  will  indicate  the  pressure  belonging  to 
it,  and  the  surface  may  be  referred  to  as  the  surface  0,  1,  2  or  3.  The  unit  of 
pressure  is  that  pressure  which  is  produced  by  unit  of  force  acting  on  unit  of 
surface.  In  order  therefore  to  diminish  the  unit  of  pressure  as  in  (5)  we  must 
diminish  the  unit  of  force  in  the  same  proportion. 

(12)  It  is  easy  to  see  that  these  surfaces  of  equal  pressure  must  be  perpen- 
dicular to  the  lines  of  fluid  motion;  for  if  the  fluid  were  to  move  in  any  other 
direction,  there  would  be  a  resistance  to  its  motion  which  could  not  be  balanced 
by  any  difference  of  pressures.  (We  must  remember  that  the  fluid  here  con- 
sidered has  no  inertia  or  mass,  and  that  its  properties  are  those  only  which  are 
formally  assigned  to  it,  so  that  the  resistances  and  pressures  are  the  only  things 


ON   Faraday's   lines  of  force.  165 

to  be  considered.)  There  are  therefore  two  sets  of  surfaces  which  by  their  inter- 
section form  the  system  of  unit  tubes,  and  the  system  of  surfaces  of  equal  pres- 
sure cuts  both  the  others  at  right  angles.  Let  h  be  the  distance  between  two 
consecutive  surfaces  of  equal  pressure  measured  along  a  line  of  motion,  then  since 
the  difference  of  pressures  =  1, 

kvh=  1, 

which  determines  the  relation  of  v  to  h,  so  that  one  can  be  found  when  the 
other  is  known.  Let  s  be  the  sectional  area  of  a  unit  tube  measured  on  a 
surface  of  equal  pressure,  then  since  by  the  definition  of  a  unit  tube 

vs  =  \, 

we  find  by  the  last  equation 

s  =  kh. 

(13)  The  surfaces  of  equal  pressure  cut  the  unit  tubes  into  portions  whose 
length  is  h  and  section  s.  These  elementary  portions  of  unit  tubes  will  be  called 
unit  cells.  In  each  of  them  unity  of  volume  of  fluid  passes  from  a  pressure  p  to 
a  pressure  (p  — 1)  in  unit  of  time,  and  therefore  overcomes  unity  of  resistance  in 
that  time.  The  work  spent  in  overcoming  resistance  is  therefore  unity  in  every 
cell  in  every  unit  of  time. 

(14)  If  the  surfaces  of  equal  pressure  are  known,  the  direction  and  magni- 
tude of  the  velocity  of  the  fluid  at  any  point  may  be  found,  after  which  the 
complete  system  of  unit  tubes  may  be  constructed,  and  the  beginnings  and  end- 
ings of  these  tubes  ascertained  and  marked  out  as  the  sources  whence  the  fluid 
is  derived,  and  the  sinks  where  it  disappears.  In  order  to  prove  the  converse  of 
this,  that  if  the  distribution  of  sources  be  given,  the  pressure  at  every  point  may 
be  found,  we  must  lay  down  certain  preliminary  propositions. 

(15)  If  we  know  the  pressures  at  every  point  in  the  fluid  in  two  different 
cases,  and  if  we  take  a  third  case  in  which  the  pressure  at  any  point  is  the 
sum  of  the  pressures  at  corresponding  points  in  the  two  former  cases,  then  the 
velocity  at  any  point  in  the  third  case  is  the  resultant  of  the  velocities  in  the 
other  two,  and  the  distribution  of  sources  is  that  due  to  the  simple  superposition 
of  the  sources  in  the  two  former  cases. 

For  the  velocity  in  any  direction  is  proportional  to  the  rate  of  decrease  of 
the   pressure    in    that   direction;   so    that    if  two    systems   of  pressures   be   added 


166  ON    FARADAY  S    LINES    OF    FORCE. 

together,  since  the  rate  of  decrease  of  pressure  along  any  line  will  be  the  sum 
of  the  combined  rates,  the  velocity  in  the  new  system  resolved  in  the  same 
direction  will  be  the  sum  of  the  resolved  parts  in  the  two  original  systems. 
The  velocity  in  the  new  system  will  therefore  be  th€  resultant  of  the  velocities 
at  corresponding  points  in  the  two  former  systems. 

It  follows  from  this,  by  (9),  that  the  (quantity  of  fluid  which  crosses  any 
fixed  surface  is,  in  the  new  system,  the  sum  of  the  corresponding  quantities  in 
the  old  ones,  and  that  the  sources  of  the  two  original  systems  are  simply 
combined  to  form  the  third. 

It  is  evident  that  in  the  system  in  which  the  pressure  is  the  diiBPerence 
of  pressure  in  the  two  given  systems  the  distribution  of  sources  will  be  got 
by  changing  the  sign  of  all  the  sources  in  the  second  system  and  adding  them 
to   those   in   the   first. 

(16)  If  the  pressure  at  every  point  of  a  closed  surface  be  the  same  and 
equal  to  p,  and  if  there  be  no  sources  or  sinks  within  the  surface,  then  there 
will  be  no  motion  of  the  fluid  within  the  surface,  and  the  pressure  within  it 
will  be  uniform  and   equal  to  p. 

For  if  there  be  motion  of  the  fluid  within  the  surface  there  will  be  tubes 
of  fluid  motion,  and  these  tubes  must  either  return  into  themselves  or  be 
terminated  either  within  the  surface  or  at  its  boundary.  Now  since  the  fluid 
always  flows  from  places  of  greater  pressure  to  places  of  less  pressure,  it 
cannot  flow  in  a  re-entering  curve;  since  there  are  no  sources  or  sinks  within 
the  surface,  the  tubes  cannot  begin  or  end  except  on  the  surface  ;  and  since 
the  pressure  at  all  points  of  the  surface  is  the  same,  there  can  be  no  motion 
in  tubes  having  both  extremities  on  the  surface.  Hence  there  is  no  motion 
within  the  surface,  and  therefore  no  difference  of  pressure  which  would  cause 
motion,  and  since  the  pressure  at  the  bounding  surface  is  p,  the  pressure  at 
any   point   within   it   is  also  p. 

(17)  If  the  pressure  at  every  point  of  a  given  closed  surface  be  known, 
and  the  distribution  of  sources  within  the  surface  be  also  known,  then  only 
one   distribution   of  pressures   can   exist   within   the   surface. 

For  if  two  different  distributions  of  pressures  satisfying  these  conditions 
could  be  found,  a  third  distribution  could  be  formed  in  which  the  pressure  at 
any  point  should  be  the  difference  of  the  pressures  in  the  two  former  distri- 
butions.    In  this  case,  since  the  pressures  at  the  surface  and  the  sources  within 


ON   Faraday's   lines   of   force.  107 

it  are  the  same  in  both  distributions,  the  pressure  at  the  surface  in  the  third 
distribution  would  be  zero,  and  all  the  sources  within  the  surface  would 
vanish,    by    (15). 

Then  by  (16)  the  pressure  at  every  point  in  the  third  distribution  must 
be  zero ;  but  this  is  the  difference  of  the  pressures  in  the  two  former  cases, 
and  therefore  these  cases  are  the  same,  and  there  is  only  one  distribution  of 
pressure   possible. 

(18)  Let  us  next  determine  the  pressure  at  any  point  of  an  infinite  body 
of  fluid  in  the  centre  of  which  a  unit  source  is  placed,  the  pressure  at  an 
infinite   distance   from   the  source   being  supposed  to   be   zero. 

The  fluid  will  flow  out  from  the  centre  symmetrically,  and  since  unity  of 
volume  flows  out  of  every  spherical  surface  surrounding  the  point  in  unit  of 
time,  the   velocity   at  a  distance  r  from  the   source   will  be 

k 

The   rate   of  decrease    of  pressure    is    therefore    hv   or  — -^,    and    since    the 

pressure  =  0    when   r  is   infinite,   the   actual   pressure   at   any   point   will  be 

=  A 

The  pressure  is  therefore  inversely  proportional  to  the  distance  from  the 
source. 

It   is    evident   that  the   pressure   due    to  a    unit    sink   will  be   negative  and 

equal  to  —  - —  . 

If  we   have   a  source   formed   by   the   coalition   of  »S'  unit   sources,   then    the 

TcS 
resulting    pressure   will    be  X>=t—,,    so    that    the    pressure    at    a    given   distance 

varies  as  the  resistance  and   number  of  sources  conjointly. 

(19)  If  a  number  of  sources  and  sinks  coexist  in  the  fluid,  then  in  order 
to  determine  the  resultant  pressure  we  have  only  to  add  the  pressures  which 
each  source  or  sink  produces.  For  by  (15)  this  will  be  a  solution  of  the 
problem,  and  by  (17)  it  will  be  the  only  one.  By  this  method  we  can 
determine   the   pressures   due  to   any    distribution    of  sources,   as   by   the   method 


168  ON  Faraday's  lines  of  forck 

of  (14)    we   can    determine   the   distribution   of  sources   to   which   a  given   distri- 
bution  of  pressures   is   due. 

(20)  We  have  next  to  shew  that  if  we  conceive  any  imaginary  surface 
as  fixed  in  space  and  intersecting  the  lines  of  motion  of  the  fluid,  we  may 
substitute  for  the  fluid  on  one  side  of  this  surface  a  distribution  of  sources 
upon  the  surface  itself  without  altering  in  any  way  the  motion  of  the  fluid 
on  the  other   side  of  the  surface. 

For  if  we  describe  the  system  of  unit  tubes  which  defines  the  motion  of 
the  fluid,  and  wherever  a  tube  enters  through  the  surface  place  a  unit  source, 
and  wherever  a  tube  goes  out  through  the  surface  place  a  unit  sink,  and  at  the 
same  time  render  the  surface  impermeable  to  the  fluid,  the  motion  of  the  fluid 
in  the  tubes  will  go   on  as  before. 

(21)  If  the  system  of  pressures  and  the  distribution  of  sources  which  pro- 
duce them  be  known  in  a  medium  whose  resistance  is  measured  by  k,  then  in 
order  to  produce  the  same  system  of  pressures  in  a  medium  whose  resistance 
is  unity,  the  rate  of  production  at  each  source  must  be  multiplied  by  k.  For 
the  pressure  at  any  point  due  to  a  given  source  varies  as  the  rate  of  produc- 
tion and  the  resistance  conjointly;  therefore  if  the  pressure  be  constant,  the 
rate  of  production  must  vary  inversely  as  the  resistance. 

(22)  On  the  conditions  to  he  fulfilled  at  a  surface  which  separates  two  media 
whose  coefficients  of  resistance  are  k  and  k\ 

These  are  found  from  the  consideration,  that  the  quantity  of  fluid  which 
flows  out  of  the  one  medium  at  any  point  flows  into  the  other,  and  that  the 
pressure  varies  continuously  from  one  medium  to  the  other.  The  velocity  normal 
to  the  surface  is  the  same  in  both  media,  and  therefore  the  rate  of  diminution 
of  pressure  is  proportional  to  the  resistance.  The  direction  of  the  tubes  of 
motion  and  the  surfaces  of  equal  pressure  will  be  altered  after  passing  through 
the  surface,  and  the  law  of  this  refraction  will  be,  that  it  takes  place  in  the 
plane  passing  through  the  direction  of  incidence  and  the  normal  to  the  surface, 
and  that  the  tangent  of  the  angle  of  incidence  is  to  the  tangent  of  the  angle 
of  refraction  as  k'  is  to  k. 

(23)  Let  the  space  within  a  given  closed  surface  be  filled  with  a  medium 
different  from  that  exterior  to  it,  and  let  the  pressures  at  any  point  of  this 
compound  system   due   to   a   given   distribution    of   sources   within    and    without 


ON  fakaday's   lines   of   force.  169 

the  surface  be  given  ;  it  is  required  to  determine  a  distribution  of  sources  which 
would  produce  the  same  system  of  pressures  in  a  medium  whose  coefficient  of 
resistance  is  unity. 

Construct  the  tubes  of  fluid  motion,  and  wherever  a  unit  tube  enters  either 
medium  place  a  unit  source,  and  wherever  it  leaves  it  place  a  unit  sink.  Then 
if  we  make  the  surface  impermeable  all  will  go  on  as  before. 

Let  the  resistance  of  the  exterior  medium  be  measured  by  k,  and  that  of 
the  interior  by  V.  Then  if  we  multiply  the  rate  of  production  of  all  the  sources 
in  the  exterior  medium  (including  those  in  the  surface),  by  k,  and  make  the 
coefficient  of  resistance  unity,  the  pressures  will  remain  as  before,  and  the  same 
will  be  true  of  the  interior  medium  if  we  multiply  all  the  sources  in  it  by  k', 
including  those  in  the  surface,  and  make  its  resistance  unity. 

Since  the  pressures  on  both  sides  of  the  surface  are  now  equal,  we  may 
suppose  it  permeable  if  we  please. 

We  have  now  the  original  system  of  pressures  produced  in  a  uniform  medium 
by  a  combination  of  three  systems  of  sources.  The  first  of  these  is  the  given 
external  system  multipHed  by  k,  the  second  is  the  given  internal  system  multi- 
plied by  k',  and  the  third  is  the  system  of  sources  and  sinks  on  the  surface 
itself.  In  the  original  case  every  source  in  the  external  medium  had  an  equal 
sink  in  the  internal  medium  on  the  other  side  of  the  surface,  but  now  the 
source  is  multiplied  by  k  and  the  sink  by  k',  so  that  the  result  is  for  every 
external  unit  source  on  the  surface,  a  source  ={k  —  k').  By  means  of  these  three 
systems  of  sources  the  original  system  of  pressures  may  be  produced  in  a  medium 
for  which  k  =  \. 

(24)  Let  there  be  no  resistance  in  the  medium  within  the  closed  surface, 
that  is,  let  /t'  =  0,  then  the  pressure  within  the  closed  surface  is  uniform  and 
equal  to  p,  and  the  pressure  at  the  surface  itself  is  also  p.  If  by  assuming 
any  distribution  of  pairs  of  sources  and  sinks  within  the  surface  in  addition  to 
the  given  external  and  internal  sources,  and  by  supposing  the  medium  the  same 
within  and  without  the  surface,  we  can  render  the  pressure  at  the  surface  uni- 
form, the  pressures  so  found  for  the  external  medium,  together  with  the  uniform 
pressure  p  in  the  internal  medium,  will  be  the  true  and  only  distribution  of 
pressures  which  is  possible. 

For   if  two  such   distributions   could  be  found  by  taking  diffijrent  imaginary 
distributions   of  pairs   of  sources   and   sinks   within  the  medium,  then  by  taking 
VOL.  I.  22 


170  ON  Faraday's  lines  of  foece. 

the  difference  of  the  two  for  a  third  distribution,  we  should  have  the  pressure 
of  the  bounding  surface  constant  in  the  new  system  and  as  many  sources  as 
sinks  within  it,  and  therefore  whatever  fluid  flows  in  at  any  point  of  the  surface, 
an  equal  quantity  must  flow  out  at  some  other  point. 

In  the  external  medium  all  the  sources  destroy  one  another,  and  we  have 
an  infinite  medium  without  sources  surrounding  the  internal  medium.  The  pres- 
sure at  infinity  is  zero,  that  at  the  surface  is  constant.  If  the  pressure  at  the 
surface  is  positive,  the  motion  of  the  fluid  must  be  outwards  from  every  point 
of  the  surface ;  if  it  be  negative,  it  must  flow  inwards  towards  the  surface.  But 
it  has  been  shewn  that  neither  of  these  cases  is  possible,  because  if  any  fluid 
enters  the  surface  an  equal  quantity  must  escape,  and  therefore  the  pressure  at 
the  surface  is  zero  in  the  third  system. 

The  pressure  at  all  points  in  the  boundary  of  the  internal  medium  in  the 
third  case  is  therefore  zero,  and  there  are  no  sources,  and  therefore  the  pressure 
is  everywhere  zero,  by  (16). 

The  pressure  in  the  bounding  surface  of  the  internal  medium  is  also  zero, 
and  there  is  no  resistance,  therefore  it  is  zero  throughout;  but  the  pressure  in 
the  third  case  is  the  difference  of  pressures  in  the  two  given  cases,  therefore 
these  are  equal,  and  there  is  only  one  distribution  of  pressure  which  is  possible, 
namely,  that  due  to  the  imaginary  distribution  of  sources  and  sinks. 

(25)  When  the  resistance  is  infinite  in  the  internal  medium,  there  can  be 
no  passage  of  fluid  through  it  or  into  it.  The  bounding  surface  may  therefore 
be  considered  as  impermeable  to  the  fluid,  and  the  tubes  of  fluid  motion  will 
run  along  it  without  cutting  it. 

If  by  assuming  any  arbitrary  distribution  of  sources  within  the  surface  in 
addition  to  the  given  sources  in  the  outer  medium,  and  by  calculating  the 
resulting  pressures  and  velocities  as  in  the  case  of  a  uniform  medium,  we  can 
fulfil  the  condition  of  there  being  no  velocity  across  the  surface,  the  system  of 
pressures  in  the  outer  medium  will  be  the  true  one.  For  since  no  fluid  passes 
through  the  surface,  the  tubes  in  the  interior  are  independent  of  those  outside, 
and  may  be  taken  away  without  altering  the  external  motion. 

(26)  If  the  extent  of  the  internal  medium  be  small,  and  if  the  difference 
of  resistance  in  the  two  media  be  also  small,  then  the  position  of  the  unit  tubes 
will  not  be  much  altered  from  what  it  would  be  if  the  external  medium  filled 
the  whole  space. 


ON     FARADAY  S    LINES    OF    FORCE.  171 

Oq  this  supposition  we  can  easily  calculate  the  kind  of  alteration  which 
the   introduction  of  the  internal  medium  will  produce ;   for  wherever  a  unit  tube 

enters   the   surface   we   must   conceive   a  source   producing   fluid   at   a  rate   -^^ , 

and   wherever   a  tube   leaves   it   we   must   place   a  sink   annihilating   fluid  at  the 

k'-k 
rate   — ^  ,    then   calculating   pressures   on   the  supposition  that   the  resistance  in 

both  media  is  k,  the  same  as  in  the  external  medium,  we  shall  obtain  the  true 
distribution  of  pressures  very  approximately,  and  we  may  get  a  better  result 
by  repeating  the  process  on  the  system  of  pressures  thus  obtained. 

(27)  If  instead  of  an  abrupt  change  from  one  coeflBcient  of  resistance  to 
another  we  take  a  case  in  which  the  resistance  varies  continuously  from  point 
to  point,  we  may  treat  the  medium  as  if  it  were  composed  of  thin  shells  each 
of  which  has  uniform  resistance.  By  properly  assuming  a  distribution  of  sources 
over  the  surfaces  of  separation  of  the  shells,  we  may  treat  the  case  as  if  the 
resistance  were  equal  to  unity  throughout,  as  in  (23).  The  sources  will  then 
be  distributed  continuously  throughout  the  whole  medium,  and  will  be  positive 
whenever  the  motion  is  from  places  of  less  to  places  of  greater  resistance,  and 
negative  when  in  the  contrary  direction. 

(28)  Hitherto  we  have  supposed  the  resistance  at  a  given  point  of  the 
medium  to  be  the  same  in  whatever  direction  the  motion  of  the  fluid  takes 
place ;  but  we  may  conceive  a  case  in  which  the  resistance  is  different  in 
different  directions.  In  such  cases  the  lines  of  motion  will  not  in  general  be 
perpendicular  to  the  surfaces  of  equal  pressure.  If  a,  6,  c  be  the  components 
of  the  velocity  at  any  point,  and  a,  yS,  y  the  components  of  the  resistance  at 
the  same  point,  these  quantities  will  be  connected  by  the  following  system  of 
linear  equations,  which  may  be  called  ''equations  of  conduction"  and  will  be 
referred  to  by  that  name. 

a^P,a  +  QS  +  R.y, 

h  =  Fj3+Q,y  +  EA, 

c  =  P,y+Q,a  +  JR,l3. 

In    these   equations   there   are   nine  independent   coefficients   of  conductivity.     In 

order  to  simplify  the  equations,  let  us  put 

Qt  +  Ji,  =  2S„     Q,-B,  =  2lT, 

&c &c. 

22—2 


172  ON  Faraday's  lines  of  force. 

where  4^  =  «?,-i2,)'  +  (^»-^.)'  +  (^3-^s)', 

and  I,  m,  n  are  direction-cosines  of  a  certain  fixed  line  in  space. 

The  equations  then  become 

a  =  P,a  +  SJ3  +  S,y  +  (nfi  -my)  T, 
b=F^  +  S,y  +  S,a  +  {lY  -  na)  T, 
c  =  P,y  +  S,a  +  S^  +  {ma~  l^)  T. 

By    the    ordinary    transformation   of  co-ordinates    we    may   get    rid    of   the 
coeflBcients  marked  S.     The  equations  then  become 

a  =  P(a  +  (n'^-m'y)T, 

b  =  P:/3  +  {ry-n'a)T, 

c  =  P,y+{m'a-  Vfi)  T, 
where  I',  m,  n'  are  the  direction-cosines   of  the   fixed  line  with  reference  to  the 
new  axes.     If  we  make 


the  equation  of  continuity 


becomes 


%^-i'  -^-|. 


da     dh      c^c  _ 
dx     dy      dz       ' 


'  dx'^    '  dy'^^'  dz'     ^' 


and  if  we  make  x  =  JP^^,     y^^fPT^],     z  =  JP^l, 

^'^^■^  3|+^  +  ?  =  °- 

the  ordinary  equation  of  conduction. 

It  appears  therefore  that  the  distribution  of  pressures  is  not  altered  by 
the  existence  of  the  coefficient  T.  Professor  Thomson  has  shewn  how  to 
conceive  a  substance  in  which  this  coefficient  determines  a  property  having 
reference  to  an  axis,  which  unlike  the  axes  of  P^,  P^,  P^  is  dipolar. 

For  further  information  on  the  equations  of  conduction,  see  Professor 
Stokes  On  the  Conduction  of  Heat  in  Crystals  {Cambridge  and  Dublin  Math. 
Journ.),  and  Professor  Thomson  On  the  Dynamical  Theory  of  Heat,  Part  v. 
{Transactions  of  Royal  Society  of  Edinburgh,  VoL  xxi.  Part  i.). 


ON  Faraday's   lines  of  force.  173 

It  is  evident  that  all  that  has  been  proved  in  (14),  (15),  (16),  (17),  with 
respect  to  the  superposition  of  different  distributions  of  pressure,  and  there  being 
only  one  distribution  of  pressures  corresponding  to  a  given  distribution  of  sources, 
will  be  true  also  in  the  case  in  which  the  resistance  varies  from  point  to  point, 
and  the  resistance  at  the  same  point  is  different  in  different  directions.  For 
il'  we  examine  the  proof  we  shall  find  it  applicable  to  such  cases  as  well  as  to 
that  of  a  uniform  medium. 

(29)  We  now  are  prepared  to  prove  certain  general  propositions  which  are 
true  in  the  most  general  case  of  a  medium  whose  resistance  is  different  in 
different  directions  and  varies  from  point  to  point. 

We  may  by  the  method  of  (28),  when  the  distribution  of  pressures  is 
known,  construct  the  surfaces  of  equal  pressure,  the  tubes  of  fluid  motion,  and 
the  sources  and  sinks.  It  is  evident  that  since  in  each  cell  into  which  a  unit 
tube  is  divided  by  the  surfaces  of  equal  pressure  unity  of  fluid  passes  from 
pressure  p  to  pressure  (p  — 1)  in  unit  of  time,  unity  of  work  is  done  by  the 
fluid  in  each  cell  in  overcoming  resistance. 

The  number  of  cells  in  each  unit  tube  is  determined  by  the  number  of 
surfaces  of  equal  pressure  through  which  it  passes.  If  the  pressure  at  the 
beginning  of  the  tube  be  p  and  at  the  end  p\,  then  the  number  of  cells  in 
it  will  be  p—p-  Now  if  the  tube  had  extended  from  the  source  to  a  place 
where  the  pressure  is  zero,  the  number  of  cells  would  have  been  p,  and  if 
the  tube  had  come  from  the  sink  to  zero,  the  number  would  have  been  p\ 
and  the  true  number  is  the  difference  of  these. 

Therefore  if  we  find  the  pressure  at  a  source  S  from  which  S  tubes 
proceed  to  be  p,  Sp  \s.  the  number  of  cells  due  to  the  source  S ;  but  if  iS'  of 
the  tubes  terminate  in  a  sink  at  a  pressure  p\  then  we  must  cut  off  S  p  cells 
from  the  number  previously  obtained.  Now  if  we  denote  the  source  of  S 
tubes  by  S,  the  sink  of  S  tubes  may  be  written  -S,  sinks  always  being 
reckoned  negative,  and  the  general  expression  for  the  number  of  cells  in  the 
system  will  be  S  (5p). 

(30)  The  same  conclusion  may  be  arrived  at  by  observing  that  unity  of 
work  is  done  on  each  cell.  Now  in  each  source  S,  S  units  of  fluid  are 
expelled  against  a  pressure  p,  so  that  the  work  done  by  the  fluid  in  over- 
coming resistance  is  Sj?.  At  each  sink  in  which  S'  tubes  terminate,  S'  units 
of  fluid   sink    into  nothing  under  pressure  p' ;   the  work  done  upon   the  fluid  by 


174  ON  Faraday's  lines  of  force. 

the  pressure  is  therefore  S' p\  The  whole  work  done  by  the  fluid  may  there- 
fore  be   expressed   by 

W  =  tSp^tS'p, 
or  more  concisely,  considering  sinks  as  negative  sources, 

W  =  t(Sp). 

(31)  Let  S  represent  the  rate  of  production  of  a  source  in  any  medium, 
and  let  p  be  the  pressure  at  any  given  point  due  to  that  source.  Then  if  we 
superpose  on  this  another  equal  source,  every  pressure  will  be  doubled,  and 
thus  by  successive  superposition  we  find  that  a  source  nS  would  produce  a 
pressure  np,  or  more  generally  the  pressure  at  any  point  due  to  a  given 
source  varies  as  the  rate  of  production  of  the  source.  This  may  be  expressed 
by  the   equation 

p  =  RS, 

where  R  is  a,  coefficient  depending  on  the  nature  of  the  medium  and  on  the 
positions  of  the  source  and  the  given  point.  In  a  uniform  medium  whose 
resistance  is  measured  by  k, 

R  may  be  called  the  coefficient  of  resistance  of  the  medium  between  the  source 
and  the  given  point.     By  combining  any  number  of  sources  we  have  generally 

p  =  %{RS), 

(32)     In  a  uniform  medium  the  pressure  due  to  a  source  S 

k    S 

At  another  source   S'  at  a  distance  r  we  shall  have 

a,        k    SS'     CI  f 

if  2^'  he  the  pressure  at  S  due  to  S\  If  therefore  there  be  two  systems  of 
sources  X{S)  and  %{S'),  and  if  the  pressures  due  to  the  first  be  p  and  to  the 
second  p',  then 

2(S»  =  2{S/). 

For  every  term  S'p  has  a  term  Sp'  equal  to  it. 


ON  Faraday's  lines  of  force.  175 

(33)  Suppose  that  in  a  uniform  medium  the  motion  of  the  fluid  is  every- 
where parallel  to  one  plane,  then  the  surfaces  of  equal  pressure  will  be 
perpendicular  to  this  plane.  If  we  take  two  parallel  planes  at  a  distance  equal 
to  k  from  each  other,  we  can  divide  the  space  between  these  planes  into  unit 
tubes  by  means  of  cylindric  surfaces  perpendicular  to  the  planes,  and  these 
together  with  the  surfaces  of  equal  pressure  will  divide  the  space  into  cells  of 
which  the  length  is  equal  to  the  breadth.  For  if  h  be  the  distance  between 
consecutive  surfaces  of  equal  pressure  and  s  the  section  of  the  unit  tube,  we 
have  by   (13)   s  =  kh. 

But  s  is  the  product  of  the  breadth  and  depth ;  but  the  depth  is  k, 
therefore   the  breadth   is   h  and  equal  to  the  length. 

If  two  systems  of  plane  curves  cut  each  other  at  right  angles  so  as  to 
divide  the  plane  into  little  areas  of  which  the  length  and  breadth  are  equal, 
then  by  taking  another  plane  at  distance  k  from  the  first  and  erecting 
cyhndric  surfaces  on  the  plane  curves  as  bases,  a  system  of  cells  will  be 
formed  which  will  satisfy  the  conditions  whether  we  suppose  the  fluid  to  run 
along  the  first  set  of  cutting  lines  or  the  second*. 


Application  of  the  Idea  of  Lines  of  Force. 

I  have  now  to  shew  how  the  idea  of  lines  of  fluid  motion  as  described 
above  may  be  modified  so  as  to  be  apphcable  to  the  sciences  of  statical  elec- 
tricity, permanent  magnetism,  magnetism  of  induction,  and  uniform  galvanic 
currents,  reserving  the  laws  of  electro-magnetism  for  special  consideration. 

I  shall  assume  that  the  phenomena  of  statical  electricity  have  been  ah*eady 
explained  by  the  mutual  action  of  two  opposite  kinds  of  matter.  If  we  consider 
one  of  these  as  positive  electricity  and  the  other  as  negative,  then  any  two 
particles  of  electricity  repel  one  another  with  a  force  which  is  measured  by  the 
product  of  the  masses  of  the  particles  divided  by  the  square  of  their  distance. 

Now  we  found  in  (18)  that  the  velocity  of  our  imaginary  fluid  due  to  a 
source  *S  at  a  distance  r  varies  inversely  as  r".  Let  us  see  what  will  be  the 
effect  of  substituting  such  a  source  for  every  particle  of  positive  electricity.  The 
velocity  due  to  each  source  would  be  proportional  to  the  attraction  due  to  the 
corresponding  particle,   and  the   resultant  velocity   due   to   all    the  sources   would 

*  See  Cambridge  and  Dublin  MalJiematical  Jownal,  Vol.  in.  p.  286. 


176  ON  Faraday's  lines  of  force. 

be  proportional  to  the  resultant  attraction  of  all  the  particles.  Now  we  may  find 
the  resultant  pressure  at  any  point  by  adding  the  pressures  due  to  the  given 
sources,  and  therefore  we  may  find  the  resultant  velocity  in  a  given  direction 
from  the  rate  of  decrease  of  pressure  in  that  direction,  and  this  will  be 
proportional  to  the  resultant  attraction  of  the  particles  resolved  in  that  direction. 
Since  the  resultant  attraction  in  the  electrical  problem  is  proportional  to 
the  decrease  of  pressure  in  the  imaginary  problem,  and  since  we  may  select 
any  values  for  the  constants  in  the  imaginary  problem,  we  may  assume  that  the 
resultant  attraction  in  any  direction  is  numerically  equal  to  the  decrease  of 
pressure  in  that  direction,   or 

ax 
By  this  assumption  we  find  that  if  F  be  the  potential, 
dV=Xdx+  Ydy  +  Zdz=  -dp, 
or  since  at  an  infinite  distance   F=  0  and  p  =  0,    V=  —p. 
In  the  electrical  problem  we  have 


7.         Q 

In  the  fluid  p  =  S  [- 


^     r 


S=  -jr  dm. 


If  k  be   supposed  very  great,  the   amount  of  fluid  produced  by   each  source 
in  order  to  keep  up  the  pressures  will  be  very  small. 

The  potential  of  any  system  of  electricity  on  itself  will  be 

If  t  (dm),  X  (dm')  be  two  systems  of  electrical  particles  and  p,  p'  the  potentials 
due  to  them  respectively,  then  by  (32) 

or  the  potential  of  the  first  system  on  the  second  is  equal  to  that  of  the  second 
system  on  the  first. 


ON  Faraday's  lines  of  force.  177 

So   that   in   the   ordinary   electrical   problems  the  analogy   in   fluid  motion  is 
of  this  kind  : 

V=-p, 

dm  =  -—  S, 
Ait 

whole  potential   of  a  system    =  -XVdm^—  W,  where   W  is  the   work  done  by 

the  fluid  in  overcoming  resistance. 

The    lines   of   forces   are  the  unit  tubes  of  fluid  motion,  and  they   may   be 
estimated  numerically  by  those  tubes. 


Theory  of  Dielectrics, 

The  electrical  induction  exercised  on  a  body  at  a  distance  depends  not 
only  on  the  distribution  of  electricity  in  the  inductric,  and  the  form  and  posi- 
tion of  the  inducteous  body,  but  on  the  nature  of  the  interposed  medium,  or 
dielectric.  Faraday*  expresses  this  by  the  conception  of  one  substance  having 
a  greater  inductive  capacity,  or  conducting  the  lines  of  inductive  action  more 
freely  than  another.  If  we  suppose  that  in  our  analogy  of  a  fluid  in  a  resisting 
medium  the  resistance  is  diflerent  in  difierent  media,  then  by  making  the 
resistance  less  we  obtain  the  analogue  to  a  dielectric  which  more  easily  conducts 
Faraday's  lines. 

It  is  evident  from  (23)  that  in  this  case  there  will  always  be  a:n  apparent 
distribution  of  electricity  on  the  surface  of  the  dielectric,  there  being  negative 
electricity  where  the  lines  enter  and  positive  electricity  where  they  emerge.  In 
the  case  of  the  fluid  there  are  no  real  sources  on  the  surface,  but  we  use 
them  merely  for  purposes  of  calculation.  In  the  dielectric  there  may  be  no 
real  charge  of  electricity,  but  only  an  apparent  electric  action  due  to  the  surface. 

If  the  dielectric  had  been  of  less  conductivity  than  the  surrounding  medium, 
we  should  have  had  precisely  opposite  eflects,  namely,  positive  electricity  where 
lines  enter,  and  negative  where  they  emerge. 


*  Series  xi. 
VOL.  I. 


23 


178  ON  Faraday's  lines  of   force. 

If  the  conduction  of  the  dielectric  is  perfect  or  nearly  so  for  the  small 
quantities  of  electricity  with  which  we  have  to  do,  then  we  have  the  case  of 
(24).  The  dielectric  is  then  considered  as  a  conductor,  its  surface  is  a  surface 
of  equal  potential,  and  the  resultant  attraction  near  the  surface  itself  is  per- 
pendicular to  it. 

Theory  of  Permanent  Magnets. 

A  magnet  is  conceived  to  be  made  up  of  elementary  magnetized  particles, 
each  of  which  has  its  own  north  and  south  poles,  the  action  of  which  upon 
other  north  and  south  poles  is  governed  by  laws  mathematically  identical  with 
those  of  electricity.  Hence  the  same  application  of  the  idea  of  lines  of  force 
can  be  made  to  this  subject,  and  the  same  analogy  of  fluid  motion  can  be 
employed  to  illustrate  it. 

But  it  may  be  useful  to  examine  the  way  in  which  the  polarity  of  the 
elements  of  a  magnet  may  be  represented  by  the  unit  cells  in  fluid  motion. 
In  each  unit  cell  unity  of  fluid  enters  by  one  face  and  flows  out  by  the  opposite 
face,  so  that  the  first  face  becomes  a  unit  sink  and  the  second  a  unit  source 
with  respect  to  the  rest  of  the  fluid.  It  may  therefore  be  compared  to  an 
elementary  magnet,  having  an  equal  quantity  of  north  and  south  magnetic 
matter  distributed  over  two  of  its  faces.  If  we  now  consider  the  cell  as  forming 
part  of  a  system,  the  fluid  flowing  out  of  one  cell  will  flow  into  the  next,  and 
so  on,  so  that  the  source  will  be  transferred  from  the  end  of  the  cell  to  the 
end  of  the  unit  tube.  If  all  the  unit  tubes  begin  and  end  on  the  bounding 
surface,  the  sources  and  sinks  will  be  distributed  entirely  on  that  surface,  and  in 
the  case  of  a  magnet  which  has  what  has  been  called  a  solenoidal  or  tubular 
distribution  of  magnetism,  all  the  imaginary  magnetic  matter  will  be  on  the 
surface^". 

Theory  of  Paramagnetic  and  Diamagnetic  Induction. 

Faraday  t  has  shewn  that  the  effects  of  paramagnetic  and  diamagnetic  bodies 
in   the    magnetic    field   may   be   explained   by    supposing    paramagnetic   bodies   to 

*  See  Professor  Thomson  On  the  Matliematical  Theory  of  Magnetism,  Chapters  in.  and  v.  Ph^. 
Trans.  1851. 

t  Experimental  Researches  (3292). 


ON     FARADAY  S     LINES    OF    FORCE.  179 

conduct  the  lines  of  force  better,  and  diamagnetic  bodies  worse,  than  the 
surrounding  medium.  Bj  referring  to  (23)  and  (26),  and  supposing  sources  to 
represent  north  magnetic  matter,  and  sinks  south  magnetic  matter,  then  if  a 
paramagnetic  body  be  in  the  neighbourhood  of  a  north  pole,  the  lines  of  force 
on  entering  it  will  produce  south  magnetic  matter,  and  on  leaving  it  they  will 
produce  an  equal  amount  of  north  magnetic  matter.  Since  the  quantities  of 
magnetic  matter  on  the  whole  are  equal,  but  the  southern  matter  is  nearest 
to  the  north  pole,  the  result  will  be  attraction.  If  on  the  other  hand  the  body 
be  diamagnetic,  or  a  worse  conductor  of  lines  of  force  than  the  surrounding 
medium,  there  will  be  an  imaginary  distribution  of  northern  magnetic  matter 
where  the  lines  pass  into  the  worse  conductor,  and  of  southern  where  they  pass 
out,  so  that  on  the  whole  there  will  be  repulsion. 

"We  may  obtain  a  more  general  law  from  the  consideration  that  the  poten- 
tial of  the  whole  system  is  proportional  to  the  amount  of  work  done  by  the 
fluid  in  overcoming  resistance.  The  introduction  of  a  second  medium  increases 
or  diminishes  the  work  done  according  as  the  resistance  is  greater  or  less  than 
that  of  the  first  medium.  The  amount  of  this  increase  or  diminution  will  vary 
as  the  square  of  the  velocity  of  the  fluid. 

Now,  by  the  theory  of  potentials,  the  moving  force  in  any  direction  is 
measured  by  the  rate  of  decrease  of  the  potential  of  the  system  in  passing  along 
that  direction,  therefore  when  ¥,  the  resistance  within  the  second  medium,  is 
greater  than  k,  the  resistance  in  the  surrounding  medium,  there  is  a  force  tend- 
ing from  places  where  the  resultant  force  v  is  greater  to  where  it  is  less,  so 
that  a  diamagnetic  body  moves  from  greater  to  less  values  of  the  resultant 
force  *. 

In  paramagnetic  bodies  V  is  less  than  k,  so  that  the  force  is  now  from 
points  of  less  to  points  of  greater  resultant  magnetic  force.  Since  these  results 
depend  only  on  the  relative  values  of  k  and  k',  it  is  evident  that  by  changing 
the  surrounding  medium,  the  behaviour  of  a  body  may  be  changed  from  para- 
magnetic to  diamagnetic  at  pleasure. 

It  is  evident  that  we  should  obtain  the  same  mathematical  results  if  we 
had  supposed  that  the  magnetic  force  had  a  power  of  exciting  a  polarity  in 
bodies   which    is   in   the   same   direction   as  the  lines  in  paramagnetic  bodies,  and 


*  Experimental   Heaearchei   (2797),   (2798).       See   Thomson,    Canibridge   and    Dublin    Mathe)naticcU 
Journal,  May,   1847. 

23—2 


180  ON  Faraday's  lines  of  force. 

in  the  reverse  direction  in  diamagnetic  bodies*.  '  In  fact  we  have  not  as  yet 
come  to  any  facts  which  would  lead  us  to  choose  any  one  out  of  these  three 
theories,  that  of  lines  of  force,  that  of  imaginary  magnetic  matter,  and  that  of 
induced  polarity.  As  the  theory  of  lines  of  force  admits  of  the  most  precise, 
and  at  the  same  time  least  theoretic  statement,  we  shall  allow  it  to  stand  for 
the  present. 

TJieory  of  Magnecrystallic  Induction. 

Ihe  theory  of  Faraday  t  with  respect  to  the  behaviour  of  crystals  in  the 
magnetic  field  may  be  thus  stated.  In  certain  crystals  and  other  substances  the 
lines  of  magnetic  force  are  conducted  with  difierent  facility  in  different  directions. 
The  body  when  suspended  in  a  uniform  magnetic  field  will  turn  or  tend  to  turn 
into  such  a  position  that  the  lines  of  force  shall  pass  through  it  with  least  resist- 
ance. It  is  not  difficult  by  means  of  the  principles  in  (28)  to  express  the  laws 
of  this  kind  of  action,  and  even  to  reduce  them  in  certain  cases  to  numerical 
formulae.  The  principles  of  induced  polarity  and  of  imaginary  magnetic  matter 
are  here  of  Httle  use;  but  the  theory  of  lines  of  force  is  capable  of  the  most 
perfect  adaptation  to  this  class  of  phenomena. 


Theory  of  the  Conduction  of  Current  Electricity. 

It  is  in  the  calculation  of  the  laws  of  constant  electric  currents  that  the 
theory  of  fluid  motion  which  we  have  laid  down  admits  of  the  most  direct  appU- 
cation.  In  addition  to  the  researches  of  Ohm  on  this  subject,  we  have  those 
of  M.  Kirchhoff,  Ann.  de  Chim.  xli.  496,  and  of  M.  Quincke,  XLvn.  203,  on  the 
Conduction  of  Electric  Currents  in  Plates.  According  to  the  received  opinions 
we  have  here  a  current  of  fluid  moving  uniformly  in  conducting  circuits,  which 
oppose  a  resistance  to  the  current  which  has  to  be  overcome  by  the  application 
of  an  electro-motive  force  at  some  part  of  the  circuit.  On  account  of  this 
resistance  to  the  motion  of  the  fluid  the  pressure  must  be  diflerent  at  difierent 
points  in  the  circuit.     This  pressure,  which  is  commonly  called  electrical  tension, 

♦  Uxp.   Ees.    (2429),   (3320).      See  Weber,   PoggendorflF,    lxxxvil   p.    H5.      Prof.   TyndaU,   Fhxi. 
Trans.  1856,  p.  237. 

t  Fxp.  Res.  (2836),  &c. 


ON    FAHADAYS    LINES    OF    FORCE.  181 

is  found  to  be  physically  identical  with  the  potential  in  statical  electricity,  and 
thus  we  have  the  means  of  connecting  the  two  sets  of  phenomena.  If  we  knew 
what  amount  of  electricity,  measured  statically,  passes  along  that  current  which 
we  assume  as  our  unit  of  current,  then  the  connexion  of  electricity  of  tension 
with  current  electricity  would  be  completed*.  This  has  as  yet  been  done  only 
approximately,  but  we  know  enough  to  be  certain  that  the  conducting  powers  of 
diflferent  substances  differ  only  in  degree,  and  that  the  difference  between  glass 
and  metal  is,  that  the  resistance  is  a  great  but  finite  quantity  in  glass,  and  a 
small  but  finite  quantity  in  metal.  Thus  the  analogy  between  statical  electricity 
and  fluid  motion  turns  out  more  perfect  than  we  might  have  supposed,  for  there 
the  induction  goes  on  by  conduction  just  as  in  current  electricity,  but  the  quan- 
tity conducted  is  insensible  owing  to  the  great  resistance  of  the  dielectricst. 


On  Electro-motive  Forces. 

When  a  uniform  current  exists  in  a  closed  circuit  it  is  evident  that  some 
other  forces  must  act  on  the  fluid  besides  the  pressures.  For  if  the  current 
were  due  to  difference  of  pressures,  then  it  would  flow  from  the  point  of 
greatest  pressure  in  both  directions  to  the  point  of  least  pressure,  whereas  in 
reahty  it  circulates  in  one  direction  constantly.  We  must  therefore  admit  the 
existence  of  certain  forces  capable  of  keeping  up  a  constant  current  in  a  closed 
circuit.  Of  these  the  most  remarkable  is  that  which  is  produced  by  chemical 
action.  A  cell  of  a  voltaic  battery,  or  rather  the  surface  of  separation  of  the 
fluid  of  the  ceU  and  the  zinc,  is  the  seat  of  an  electro-motive  force  which 
can  maintain  a  current  in  opposition  to  the  resistance  of  the  circuit.  If  we 
adopt  the  usual  convention  in  speaking  of  electric  currents,  the  positive  current 
is  from  the  fluid  through  the  platinum,  the  conducting  circuit,  and  the  zinc, 
back  to  the  fluid  again.  If  the  electro-motive  force  act  only  in  the  surface  of 
separation  of  the  fluid  and  zinc,  then  the  tension  of  electricity  in  the  fluid 
must  exceed  that  in  the  zinc  by  a  quantity  depending  on  the  nature  and 
length  of  the  circuit  and  on  the  strength  of  the  current  in  the  conductor. 
In  order  to  keep  up  this  difference  of  pressure  there  must  be  an  electro-motive 
force  whose  intensity  is  measured  by  that  difference  of  pressure.  If  F  be  the 
electro-motive  force,  /  the   quantity   of  the   current   or   the   number   of  electrical 

♦  See  Exp.  Ees.  (371).  t  Hxp.  Ret.  Vol  iii.  p.  513. 


182  ON  Faraday's  lines  of  force. 

units   delivered   in   unit   of  time,   and   K  a    quEfntity    depending  on    the  length 
and  resistance  of  the  conducting   circuit,  then 

F=IK=p-p\ 
where  p   is   the   electric   tension   in  the  fluid  and  p'  in  the  zinc. 

If  the  circuit  be  broken  at  any  point,  then  since  there  is  no  current  the 
tension  of  the  part  which  remains  attached  to  the  platinum  will  be  p,  and 
that  of  the  other  will  be  p,  p-p  or  F  afibrds  a  measure  of  the  intensity 
of  the  current.  This  distinction  of  quantity  and  intensity  is  very  useful  *, 
but  must  be  distinctly  understood  to  mean  nothing  more  than  this : — The 
quantity  of  a  current  is  the  amount  of  electricity  which  it  transmits  in  unit 
of  time,  and  is  measured  by  /  the  number  of  unit  currents  which  it  contains. 
The  intensity  of  a  current  is  its  power  of  overcoming  resistance,  and  is 
measured  by   F  or   IK,  where   K  is   the   resistance   of  the   wliole  circuit. 

The   same   idea    of  quantity   and   intensity    may   be   applied   to   the  case   of 

magnetism  f.      The    quantity    of    magnetization    in    any    section    of    a    magnetic 

body  is  measured  by  the  number  of  lines  of  magnetic   force  which  pass  through 

it.     The    intensity    of    magnetization    in   the    section   depends    on    the    resisting 

power   of   the   section,   as   well   as   on   the   number    of  lines   which  pass   through 

it.     If  h  be  the  resisting  power  of  the  material,   and   S  the   area   of  the  section, 

and  /  the   number   of  lines   of  force   which    pass    through    it,    then    the    whole 

intensity   throughout  the   section 

h 
=  F=I- 

When  magnetization  is  produced  by  the  influence  of  other  magnets  only, 
we  may  put  p  for  the  magnetic  tension  at  any  point,  then  for  the  whole 
magnetic  solenoid 


F=l(^dx  =  IK=p-p, 


When  a  solenoidal  magnetized  circuit  returns  into  itself,  the  magnetization 
does  not  depend  on  difference  of  tensions  only,  but  on  some  magnetizing  force 
of  which   the  intensity   is   F. 

If  i  be  the  quantity  of  the  magnetization  at  any  point,  or  the  number  of 
lines   of  force  passing  through  unit   of  area  in   the   section  of  the  solenoid,   then 

*  Hxp.  Res.  Vol.  HI.  p.  519.  t  Exp.  Res.  (2870),  (3293). 


ON   Faraday's  lines  of  force.  183 

the  total  quantity  of  magnetization  in  the  circuit  is  the  number  of  lines  which 
pass  through  any  section,  I=Xidydz,  where  dydz  is  the  element  of  the  section, 
and  the  summation   is  performed  over   the   whole   section. 

The  intensity  of  magnetization  at  any  point,  or  the  force  required  to 
keep  up  the  magnetization,  is  measured  by  Jci=f,  and  the  total  intensity  of 
magnetization  in  the  circuit  is  measured  by  the  sum  of  the  local  intensities  all 
round  the  circuit, 

F=t(fdx), 

where  dx  is  the  element  of  length  in  the  circuit,  and  the  summation  is  extended 
round  the  entire   circuit. 

In  the  same  circuit  we  have  always  F  =  IK,  where  K  is  the  total  resistance 
of  the  circuit,  and  depends  on  its  form  and  the  matter  of  which  it  is 
composed. 


On  the  Action  of  closed  Currents  at  a   Distance. 

The  mathematical  laws  of  the  attractions  and  repulsions  of  conductors  have 
been  most  ably  investigated  by  Ampere,  and  his  results  have  stood  the  test  of 
subsequent  experiments. 

From  the  single  assumption,  that  the  action  of  an  element  of  one  current 
upon  an  element  of  another  current  is  an  attractive  or  repulsive  force  acting 
in  the  direction  of  the  line  joining  the  two  elements,  he  has  determined  by 
the  simplest  experiments  the  mathematical  form  of  the  law  of  attraction,  and 
has  put  this  law  into  several  most  elegant  and  useful  forms.  We  must 
recollect  however  that  no  experiments  have  been  made  on  these  elements  of 
currents  except  under  the  form  of  closed  currents  either  in  rigid  conductors 
or  in  fluids,  and  that  the  laws  of  closed  currents  can  only  be  deduced  from 
such  experiments.  Hence  if  Ampere's  formulae  applied  to  closed  currents  give 
true  results,  their  truth  is  not  proved  for  elements  of  currents  unless  we 
assume  that  the  action  between  two  such  elements  must  be  along  the  line  which 
joms  them.  Although  this  assumption  is  most  warrantable  and  philosophical  in 
the  present  state  of  science,  it  wiQ  be  more  conducive  to  freedom  of  investi- 
gation if  we  endeavour  to  do  without  it,  and  to  assume  the  laws  of  closed  currents 
as  the  ultimate  datum  of  experiment. 


384  ON  fahaday's  lines  of  force. 

Ampere  has  shewn  that  when  currents  are  combined  according  to  the  law 
of  the  parallelogram  of  forces,  the  force  due  to  the  resultant  current  is  the 
resultant  of  the  forces  due  to  the  component  currents,  and  that  equal  and 
opposite  currents  generate  equal  and  opposite  forces,  and  when  combined 
neutralize  each  other. 

He  has  also  shewn  that  a  closed  circuit  of  any  form  has  no  tendency  to 
turn  a  moveable  circular  conductor  about  a  fixed  axis  through  the  centre  of 
the  circle  perpendicular  to  its  plane,  and  that  therefore  the  forces  in  the  case 
of  a  closed  circuit  render  Xdx  +  Ydy  +  Zdz  a  complete  differential. 

Finally,  he  has  shewn  that  if  there  be  two  systems  of  circuits  similar 
and  similarly  situated,  the  quantity  of  electrical  current  in  corresponding 
conductors  being  the  same,  the  resultant  forces  are  equal,  whatever  be  the 
absolute  dimensions  of  the  systems,  which  proves  that  the  forces  are,  cceteris 
paribus,  inversely  as  the  square  of  the  distance. 

From  these  results  it  follows  that  the  mutual  action  of  two  closed  currents 
whose  areas  are  very  small  is  the  same  as  that  of  two  elementary  magnetic 
bars  magnetized  perpendicularly  to  the  plane  of  the  currents. 

The  direction  of  magnetization  of  the  equivalent  magnet  may  be  pre- 
dicted by  remembering  that  a  current  travelling  round  the  earth  from  east 
to  west  as  the  sun  appears  to  do,  would  be  equivalent  to  that  magnetization 
which  the  earth  actually  possesses,  and  therefore  in  the  reverse  direction  to 
that  of  a  magnetic  needle  when  pointing  freely. 

If  a  number  of  closed  unit  currents  in  contact  exist  on  a  surface,  then  at 
aU  points  in  which  two  currents  are  in  contact  there  will  be  two  equal  and 
opposite  currents  which  will  produce  no  effect,  but  all  round  the  boundary  of  the 
surfeice  occupied  by  the  currents  there  will  be  a  residual  current  not  neutralized 
by  any  other;  and  therefore  the  result  will  be  the  same  as  that  of  a  single 
unit   current  round  the  boundary  of  all  the   currents. 

From  this  it  appears  that  the  external  attractions  of  a  shell  uniformly 
magnetized  perpendicular  to  its  surface  are  the  same  as  those  due  to  a  current 
round  its  edge,  for  each  of  the  elementary  currents  in  the  former  case  has 
the  same  effect  as  an  element  of  the  magnetic  shell. 

If  we  examine  the  Unes  of  magnetic  force  produced  by  a  closed  current, 
we  shall  find  that  they  form  closed  curves  passing  round  the  current  and 
embracing  it,  and  that  the  total  intensity  of  the  magnetizing  force  all  along 
the   closed  line   of  force   depends   on   the  quantity  of  the  electric    current  only. 


ON    FARADAY  3    LINES    OF    FORCE.  185 

The  number  of  unit  lines*  of  magnetic  force  due  to  a  closed  current  depends 
on  the  form  as  well  as  the  quantity  of  the  current,  but  the  number  of  unit 
cells t  in  each  complete  line  of  force  is  measured  simply  by  the  number  of  unit 
currents  which  embrace  it.  The  unit  cells  in  this  case  are  portions  of  space  in 
which  unit  of  magnetic  quantity  is  produced  by  unity  of  magnetizing  force. 
The  length  of  a  cell  is  therefore  inversely  as  the  intensity  of  the  magnetizing 
force,  and  its  section  inversely  as  the  quantity  of  magnetic  induction  at  that 
point. 

The  whole  number  of  cells  due  to  a  given  current  is  therefore  proportional 
to  the  strength  of  the  current  multiplied  by  the  number  of  lines  of  force 
which  pass  through  it.  If  by  any  change  of  the  form  of  the  conductors  the 
number  of  cells  can  be  increased,  there  will  be  a  force  tending  to  produce  that 
change,  so  that  there  is  always  a  force  urging  a  conductor  transverse  to  the 
lines  of  magnetic  force,  so  as  to  cause  more  lines  of  force  to  pass  throuo-h  the 
closed   circuit   of   which  the   conductor   forms   a    part. 

The  number  of  cells  due  to  two  given  currents  is  got  by  multiplying 
the  number  of  lines  of  inductive  magnetic  action  which  pass  through  each  by 
the  quantity  of  the  currents  respectively.  Now  by  (9)  the  number  of  lines 
which  pass  through  the  first  current  is  the  sum  of  its  own  lines  and  those 
of  the  second  current  which  would  pass  through  the  first  if  the  second  current 
alone  were  in  action.  Hence  the  whole  number  of  cells  will  be  increased  by 
any  motion  which  causes  more  lines  of  force  to  pass  through  either  circuit, 
and  therefore  the  resultant  force  will  tend  to  produce  such  a  motion,  and  the 
work  done  by  this  force  during  the  motion  will  be  measured  by  the  number 
of  new  cells  produced.  All  the  actions  of  closed  conductors  on  each  other  may 
be  deduced  from  this  principle. 

On  Electric  Currents  prodiiced  by  Induction. 

Faraday  has  shewn|  that  when  a  conductor  moves  transversely  to  the  lines 
of  magnetic  force,  an  electro-motive  force  arises  in  the  conductor,  tending  to 
produce  a  current  in  it.  If  the  conductor  is  closed,  there  is  a  continuous 
current,  if  open,  tension  is  the  result.  If  a  closed  conductor  move  transversely 
to   the   lines    of  magnetic  induction,    then,    if  the    number    of   lines    which   pass 

♦  Hxp.  Rea.  (3122).     See  Art.  (6)  of  this  paper.  t  Art.  (13). 

X  Exp.  lies.  (3077),  &c. 

VOL.  I.  24 


186  ON  Faraday's   lines  of   force. 

through  it  does  not  change  during  the  motion,  the  electro-motive  forces  in  the 
circuit  will  be  in  equilibrium,  and  there  will  be  no  current.  Hence  the  electro- 
motive forces  depend  on  the  number  of  lines  which  are  cut  by  the  conductor 
during  the  motion.  If  the  motion  be  such  that  a  greater  number  of  lines  pass 
through  the  circuit  formed  by  the  conductor  after  than  before  the  motion, 
then  the  electro-motive  force  will  be  measured  by  the  increase  of  the  number 
of  lines,  and  will  generate  a  current  the  reverse  of  that  which  would  have 
produced  the  additional  Hnes.  When  the  number  of  lines  of  inductive  magnetic 
action  through  the  circuit  is  increased,  the  induced  current  will  tend  to  diminish 
the  number  of  lines,  and  when  the  number  is  diminished  the  induced  current 
will  tend  to  increase  them. 

That  this  is  the  true  expression  for  the  law  of  induced  currents  is  shewn 
from  the  fact  that,  in  whatever  way  the  number  of  lines  of  magnetic  induction 
passing  through  the  circuit  be  increased,  the  electro-motive  effect  is  the  same, 
whether  the  increase  take  place  by  the  motion  of  the  conductor  itself,  or  of  other 
conductors,  or  of  magnets,  or  by  the  change  of  intensity  of  other  currents,  or 
by  the  magnetization  or  demagnetization  of  neighbouring  magnetic  bodies,  or 
lastly  by  the  change  of  intensity  of  the   current  itself. 

In  all  these  cases  the  electro-motive  force  depends  on  the  change  in  the 
number  of  lines  of  inductive  magnetic   action  which  pass  through  the  circuit*. 

*  The  electro-magnetic  forces,  which  tend  to  produce  motion  of  the  material  conductor,  must  be 
carefully  distinguished  from  the  electro-motive  forces,  which  tend  to  produce  electric  currents. 

Let  an  electric  current  be  passed  through  a  mass  of  metal  of  any  form.  The  distribution  of 
the  currents  within  the  metal  will  be  determined  by  the  laws  of  conduction.  Now  let  a  constant 
electric  cuiTent  be  passed  through  another  conductor  near  the  first.  If  the  two  currents  are  in  the 
same  direction  the  two  conductors  will  be  attracted  towards  each  other,  and  would  come  nearer  if 
not  held  in  their  positions.  But  though  the  material  conductors  are  attracted,  the  currents  (which 
are  free  to  choose  any  course  within  the  metal)  will  not  alter  their  original  distribution,  or  incline 
towards  each  other.  For,  since  no  change  takes  place  in  the  system,  there  will  be  no  electro-motive 
forces  to  modify  the  original  distribution  of  currents. 

In  this  case  we  have  electro-magnetic  forces  acting  on  the  material  conductor,  without  any 
electi"o-motive  forces  tending  to  modify  the  current  which  it  can-ies. 

Let  us  take  as  another  example  the  case  of  a  linear  conductor,  not  forming  a  closed  circuit, 
and  let  it  be  made  to  traverse  the  lines  of  magnetic  force,  either  by  its  own  motion,  or  by  changes 
in  the  magnetic  field.  An  electro-motive  force  wiU  act  in  the  direction  of  the  conductor,  and,  as  it 
cannot  produce  a  current,  because  there  is  no  circuit,  it  will  produce  electric  tension  at  the  extremi- 
ties. There  will  be  no  electro-magnetic  attraction  on  the  material  conductor,  for  this  attraction 
depends  on  the  existence  of  the  cun-ent  within  it,  and  this  is  prevented  by  the  circuit  not  being  closed. 

Here  then  we  have  the  opposite  case  of  an  electro-motive  force  acting  on  the  electricity  in  the 
conductor,  but  no  attraction  on  its  material  particles. 


ON     FARADAY  8    LINES    OF    FORCE.  187 

It  is  natural  to  suppose  that  a  force  of  this  kind,  which  depends  on  a 
change  in  the  number  of  lines,  is  due  to  a  change  of  state  which  is  measured 
by  the  number  of  these  lines.  A  closed  conductor  in  a  magnetic  field  may- 
be supposed  to  be  in  a  certain  state  arising  from  the  magnetic  action. 
As  long  as  this  state  remains  unchanged  no  effect  takes  place,  but,  when  the 
state  changes,  electro-motive  forces  arise,  depending  as  to  their  intensity  and 
direction  on  this  change  of  state.  I  cannot  do  better  here  than  quote  a 
passage   from  the  first  series  of  Faraday's  Experimental  Researches,    Art.    (60). 

"While  the  wire  is  subject  to  either  volta-electric  or  magno-electric 
induction  it  appears  to  be  in  a  peculiar  state,  for  it  resists  the  formation  of 
an  electrical  current  in  it ;  whereas,  if  in  its  common  condition,  such  a  current 
would  be  produced;  and  when  left  uninfluenced  it  has  the  power  of  originating  a 
current,  a  power  which  the  wire  does  not  possess  under  ordinary  circumstances. 
This  electrical  condition  of  matter  has  not  hitherto  been  recognised,  but  it 
probably  exerts  a  very  important  influence  in  many  if  not  most  of  the  phe- 
nomena produced  by  currents  of  electricity.  For  reasons  which  will  immediately 
appear  (7)  I  have,  after  advising  with  several  learned  friends,  ventured  to 
designate  it  as  the  electro-tonic  state."  Finding  that  all  the  phenomena  could 
be  otherwise  explained  without  reference  to  the  electro-tonic  state,  Faraday  in 
his  second  series  rejected  it  as  not  necessary ;  but  in  his  recent  researches  ■'"' 
he  seems  still  to  think  that  there  may  be  some  physical  truth  in  his 
conjecture  about  this  new  state  of  bodies. 

The  conjecture  of  a  philosopher  so  familiar  with  nature  may  sometimes  be 
more  pregnant  with  truth  than  the  best  established  experimental  law  disco- 
vered by  empirical  inquirers,  and  though  not  bound  to  admit  it  as  a  physical 
truth,  we  may  accept  it  as  a  new  idea  by  which  our  mathematical  conceptions 
may  be  rendered  clearer. 

In  this  outline  of  Faraday's  electrical  theories,  as  they  appear  from  a 
mathematical  point  of  view,  I  can  do  no  more  than  simply  state  the  mathe- 
matical methods  by  which  I  believe  that  electrical  phenomena  can  be  best 
comprehended  and  reduced  to  calculation,  and  my  aim  has  been  to  present  the 
mathematical  ideas  to  the  mind  in  an  embodied  form,  as  systems  of  lines  or 
surfaces,  and  not  as  mere  symbols,  which  neither  convey  the  same  ideas,  nor 
readily  adapt  themselves  to  the  phenomena  to  be  explained.  The  idea  of  the 
electro-tonic  state,    however,    has    not  yet   presented   itself  to  my  mind  in  such  a 

*  (3172)  (3269). 

24—2 


188  ON  Faraday's  lines  of  force. 

form  that  its  nature  and  properties  may  be  clearly  explained  witliout  reference 
to  mere  symbols,  and  therefore  I  propose  in  the  following  investigation  to  use 
symbols  freely,  and  to  take  for  granted  the  ordinary  mathematical  operations. 
By  a  careful  study  of  the  laws  of  elastic  solids  and  of  the  motions  of  viscous 
fluids,  I  hope  to  discover  a  method  of  forming  a  mechanical  conception  of  this 
electro- tonic  state  adapted  to  general  reasoning*. 


Part  II. 

On  Faraday's   " Electro^tonic  State" 

When  a  conductor  moves  in  the  neighbourhood  of  a  current  of  electricity, 
or  of  a  magnet,  or  when  a  current  or  magnet  near  the  conductor  is  moved,  or 
altered  in  intensity,  then  a  force  acts  on  the  conductor  and  produces  electric 
tension,  or  a  continuous  current,  according  as  the  circuit  is  open  or  closed.  This 
current  is  produced  only  by  changes  of  the  electric  or  magnetic  phenomena  sur- 
rounding the  conductor,  and  as  long  as  these  are  constant  there  is  no  observed 
effect  on  the  conductor.  Still  the  conductor  is  in  different  states  when  near-  a 
current  or  magnet,  and  when  away  from  its  influence,  since  the  removal  or 
destruction  of  the  current  or  magnet  occasions  a  current,  which  would  not  have 
existed  if  the   magnet   or   current  had  not  been  previously  in  action. 

Considerations    of    this    kind    led    Professor    Faraday   to   connect    with    his 

discovery  of  the  induction  of  electric  currents  the  conception  of  a  state  into 
which  all  bodies  are  thrown  by  the  presence  of  magnets  and  currents.  This 
state  does  not  manifest  itself  by  any  known  phenomena  as  long  as  it  is  undis- 
turbed, but  any  change  in  this  state  is  indicated  by  a  current  or  tendency 
towards  a  current.  To  this  state  he  gave  the  name  of  the  "  Electro-tonic 
State,"  and  although  he  afterwards  succeeded  in  explaining  the  phenomena 
which  suggested  it  by  means  of  less  hypothetical  conceptions,  he  has  on  several 
occasions  hinted  at  the  probability  that  some  phenomena  might  be  discovered 
which  would  render  the  electro-tonic  state  an  object  of  legitimate  induction. 
These  speculations,  into  which  Faraday  had  been  led  by  the  study  of  laws 
which  he  has  well  established,  and  which  he  abandoned  only  for  want  of  experi- 

*  See   Pro£    W.   Thomson   On  a  Mechanical  Representation   of  Electric,   Magnetic    and   Galvanic 
Forces.     Camvb.  and  Dub.  Math.  Jour.     Jan.  1847. 


ON   f^vraday's  lines  of  forcr  189 

mental  data  for  the  direct  proof  of  the  unknown  state,  have  not,  I  think,  been 
made  the  subject  of  mathematical  investigation.  Perhaps  it  may  be  thought 
that  the  quantitative  determinations  of  the  various  phenomena  are  not  suffi- 
ciently rigorous  to  be  made  the  basis  of  a  mathematical  theory ;  Faraday, 
however,  has  not  contented  himself  with  simply  stating  the  numerical  results  of 
his  experiments  and  leaving  the  law  to  be  discovered  by  calculation.  Where 
he  has  perceived  a  law  he  has  at  once  stated  it,  in  terms  as  unambiguous  as 
those  of  pure  mathematics ;  and  if  the  mathematician,  receiving  this  as  a  physical 
truth,  deduces  from  it  other  laws  capable  of  being  tested  by  experiment,  he 
has  merely  assisted  the  physicist  in  arranging  his  own  ideas,  which  is  con- 
fessedly  a   necessary   step   in   scientific  induction. 

In  the  following  investigation,  therefore,  the  laws  established  by  Faraday 
will  be  assumed  aa  true,  and  it  will  be  shewn  that  by  following  out  his 
speculations  other  and  more  general  laws  can  be  deduced  from  them.  If  it 
should  then  appear  that  these  laws,  originally  devised  to  include  one  set  of 
phenomena,  may  be  generalized  so  as  to  extend  to  phenomena  of  a  different 
class,  these  mathematical  connexions  may  suggest  to  physicists  the  means  of 
establishing  physical  connexions;  and  thus  mere  speculation  may  be  turned  to 
account   in   experimental  science. 

On   Quantity  and  Intensity  as  Properties   of  Electric    Currents. 

It  is  found  that  certain  effects  of  an  electric  current  are  equal  at  what- 
ever part  of  the  circuit  they  are  estimated.  The  quantities  of  water  or  of 
any  other  electrolyte  decomposed  at  two  different  sections  of  the  same  circuit, 
are  always  found  to  be  equal  or  equivalent,  however  different  the  material  and 
form  of  the  circuit  may  be  at  the  two  sections.  The  magnetic  effect  of  a 
conducting  wire  is  also  found  to  be  independent  of  the  form  or  material  of 
the  wire  in  the  same  circuit.  There  is  therefore  an  electrical  effect  which  is 
equal  at  every  section  of  the  circuit.  If  we  conceive  of  the  conductor  as  the 
channel  along  which  a  fluid  is  constrained  to  move,  then  the  quantity  of  fluid 
transmitted  by  each  section  will  be  the  same,  and  we  may  define  the  quantity 
of  an  electric  current  to  be  the  quantity  of  electricity  which  passes  across  a 
complete  section  of  the  current  in  unit  of  time.  We  may  for  the  present 
measure  quantity  of  electricity  by  the  quantity  of  water  which  it  would  decom- 
pose in  unit  of  time. 


190  ON    FABADAYS    LINES    OF    FORCE. 

In  order  to  express  mathematically  the  electrical  currents  in  any  conductor, 
we  must  have  a  definition,  not  only  of  the  entire  flow  across  a  complete  section, 
but  also  of  the  flow  at  a  given  point  in  a  given  direction. 

Def.  The  quantity  of  a  current  at  a  given  point  and  in  a  given  direction 
is  measured,  when  uniform,  by  the  quantity  of  electricity  which  flows  across 
unit  of  area  taken  at  that  point  perpendicular  to  the  given  direction,  and  when 
variable  by  the  quantity  which  would  flow  across  this  area,  supposing  the  flow 
uniformly  the  same  as  at  the  given  point. 

In  the  following  investigation,  the  quantity  of  electric  current  at  the  point 
(xyz)  estimated  in  the  directions  of  the  axes  x,  y,  z  respectively  will  be  denoted 
by  Oj,  5j,  C3. 

The  quantity  of  electricity  which  flows  in  unit  of  time  through  the  ele- 
mentary area  dS 

=  dS  (la^  +  ?nZ)2  +  nc^), 
where  I,  m,  n  are  the  direction-cosines  of  the  normal  to  dS. 

This  flow  of  electricity  at  any  point  of  a  conductor  is  due  to  the  electro- 
motive forces  which  act  at  that  point.     These  may  be  either  external  or  internal. 

External  electro- motive  forces  arise  either  from  the  relative  motion  of  currents 
and  magnets,  or  from  changes  in  their  intensity,  or  from  other  causes  acting 
at   a   distance. 

Internal  electro-motive  forces  arise  principally  from  diSerence  of  electric 
tension  at  points  of  the  conductor  in  the  immediate  neighbourhood  of  the  point 
in  question.  The  other  causes  are  variations  of  chemical  composition  or  of  tem- 
perature in  contiguous  parts  of  the  conductor. 

Let  Pi  represent  the  electric  tension  at  any  point,  and  X^,  F,,  Z,  the  sums 
of  the  parts  of  all  the  electro-motive  forces  arising  from  other  causes  resolved 
parallel  to  the  co-ordinate  axes,  then  if  Og,  ySj,  y^  be  the  efiective  electro-motive 
forces 

"^-^^'dx 


dp, 
^'-^'"dy 

dp, 

y^^^'^-d^ 


(A). 


ON  Faraday's  lines  of  force.  191 

Now  the  quantity  of  the  current  depends  on  the  electro-motive  force  and 
on  the  resistance  of  the  medium.  If  the  resistance  of  the  medium  be  uniform 
in  all  directions  and  equal  to  k^, 

a^  =  Jc,a„         ^,  =  kK         y2  =  Kc2 (B), 

but  if  the  resistance  be  different  in  different  directions,  the  law  will  be  more 
complicated. 

These  quantities  Oj,  /3j,  y.,  may  be  considered  as  representing  the  intensity 
of  the  electric  action  in  the  directions  of  x,  y,  z. 

The  intensity  measured  along  an  element  da  of  a  curve  is  given  by 

€  =  Za  +  mji  +  ny, 
where  Z,  m,  n  are  the  direction-cosines  of  the  tangent. 

The  integral  JecZcr  taken  with  respect  to  a  given  portion  of  a  curve  line, 
represents  the  total  intensity  along  that  line.  If  the  curve  is  a  closed  one,  it 
represents  the  total  intensity  of  the  electro-motive  force  in  the  closed  curve. 

Substituting  the  values  of  a,  /8,  y  from  equations  (A) 

l^da-  =  l{Xdx  +  Ydy  +  Zdz)  -p  +  a 

If  therefore  {Xdx+  Ydy  +  Zdz)  is  a  complete  differential,  the  value  of  Jedo-  for 
a  closed  curve  will  vanish,  and  in  all  closed  curves 

leda-  =  l{Xdx+Ydy  +  Zdz), 

the  integration  being  effected  along  the  curve,  so  that  in  a  closed  curve  the 
total  intensity  of  the  effective  electro- motive  force  is  equal  to  the  total  intensity 
of  the  impressed  electro-motive  force. 

The  total  quantity  of  conduction  through  any  surface  is  expressed  by 

\edS, 
where 

e  =  la  +  mh  +  nc, 

I,  m,  n  being  the  direction- cosines  of  the  normal, 

.  •.    \edS  =  l\adydz  +  ^bdzdx  +  \\cdxdy, 

the  integrations  being  effected  over  the  given  surface.  AVhen  the  surface  is  a 
closed  one,  then  we  may  find  by  integration  by  parts 


w.=///(:- 


7a     dh      dc\  ,    ,    , 


192  ON    FARADAY  S    LINES    OF    FORCE. 

If  we  make 

da     dh   ^  d.c  /^v 

Tx  +  dy+di^^^P (^)' 

\edS=  iirlWpdxdydz, 

where  the  integration  on  the  right  side  of  the  equation  is  effected  over  every 
part  of  space  within  the  surface.  In  a  large  class  of  phenomena,  including  all 
cases  of  uniform  currents,  the  quantity  p  disappears. 


Magnetic  Quantity  and  Intensity. 

From  his  study  of  the  lines  of  magnetic  force,  Faraday  has  been  led  to 
the  conclusion  that  in  the  tubular  surface  ■'''  formed  by  a  system  of  such  lines, 
the  quantity  of  magnetic  induction  across  any  section  of  the  tube  is  constant, 
and  that  the  alteration  of  the  character  of  these  lines  in  passing  from  one 
substance  to  another,  is  to  be  explained  by  a  difference  of  inductive  capacity 
in  the  two  substances,  which  is  analogous  to  conductive  power  in  the  theory 
of  electric  currents. 

In  the  following  investigation  we  shall  have  occasion  to  treat  of  magnetic 
quantity  and  intensity  in  connection  with  electric.  In  such  cases  the  magnetic 
symbols  wiU  be  distinguished  by  the  sufiix  1,  and  the  electric  by  the  suffix  2. 
The  equations  connecting  a,  h,  c,  h,  a,  /8,  y,  p,  and  p,  are  the  same  in  form  as 
those  which  we  have  just  given,  a,  6,  c  are  the  symbols  of  magnetic  induction 
with  respect  to  quantity ;  k  denotes  the  resistance  to  magnetic  induction,  and 
may  be  different  in  different  directions ;  a,  /8,  y,  are  the  effective  magnetiang 
forces,  connected  with  a,  h,  c,  by  equations  (B) ;  p  is  the  magnetic  tension  or 
potential  which  will  be  afterwards  explained ;  p  denotes  the  density  of  real 
magnetic  matter  and  is  connected  with  a,  h,  c  by  equations  (C).  As  all  the 
details  of  magnetic  calculations  will  be  more  intelligible  after  the  exposition  of  the 
connexion  of  magnetism  with  electricity,  it  will  be  sufficient  here  to  say  that 
all  the  definitions  of  total  quantity,  with  respect  to  a  surface,  the  total  intensity 
to  a  curve,  apply  to  the  case  of  magnetism  as  well  as  to  that  of  electricity. 

*  Exp.  Res.  3271,  definition  of  "  Sphondyloid." 


ON   Faraday's   lines  of  force.  193 


Electro-magnetism. 


Ampere  has  proved  the  following  laws  of  the  attractions  and  repulsions  of 
electric  currents  : 

I.  Equal  and  opposite  currents  generate  equal  and  opposite  forces. 

II.  A  crooked  current  is  equivalent  to  a  straight  one,  provided  the  two 
currents  nearly  coincide  throughout  their  whole  length. 

IIL  Equal  currents  traversing  similar  and  similarly  situated  closed  curves 
act  with  equal  forces,  whatever  be  the  linear  dimensions  of  the  circuits. 

IV.  A  closed  current  exerts  no  force  tending  to  turn  a  circular  conductor 
about  its  centre. 

It  is  to  be  observed,  that  the  currents  with  which  Ampere  worked  were  constant 
and  therefore  re-entering.  All  his  results  are  therefore  deduced  from  experiments 
on  closed  currents,  and  his  expressions  for  the  mutual  action  of  the  elements 
of  a  current  involve  the  assumption  that  this  action  is  exerted  in  the  direction 
of  the  line  joining  those  elements.  This  assumption  is  no  doubt  warranted  by  the 
universal  consent  of  men  of  science  in  treating  of  attractive  forces  considered 
as  due  to  the  mutual  action  of  particles ;  but  at  present  we  are  proceeding 
on  a  different  principle,  and  searching  for  the  explanation  of  the  phenomena, 
not  in  the  currents  alone,  but  also  in  the  surrounding  medium. 

The  first  and  second  laws  shew  that  currents  are  to  be  combined  like 
velocities  or  forces. 

The  third  law  is  the  expression  of  a  property  of  all  attractions  which  may 
be  conceived  of  as  depending  on  the  inverse  square  of  the  distance  from  a  fixed 
system  of  points ;  and  the  fourth  shews  that  the  electro-magnetic  forces  may 
always  be  reduced  to  the  attractions  and  repulsions  of  imaginary  matter  properly 
distributed. 

In  fact,  the  action  of  a  very  small  electric  circuit  on  a  point  in  its  neigh- 
bourhood is  identical  with  that  of  a  small  magnetic  element  on  a  point  outside 
it.  If  we  divide  any  given  portion  of  a  surface  into  elementary  areas,  and 
cause  equal  currents  to  flow  in  the  same  direction  round  all  these  Httle  areas, 
the  effect  on  a  point  not  in  the  surface  will  be  the  same  as  that  of  a  shell 
coinciding  with  the  surface,  and  uniformly  magnetized  normal  to  its  surface. 
But    by   the   first   law   all    the   currents   forming   the    little    circuits   will   destroy 

VOL.  L  25 


194  ON    FARADAY  S    LINES    OF    FORCE. 

one  another,  and  leave  a  single  current  running  round  the  bounding  line.  So 
that  the  magnetic  effect  of  a  uniformly  magnetized  shell  is  equivalent  to  that 
of  an  electric  current  round  the  edge  of  the  shell.  If  the  direction  of  the  current 
coincide  with  that  of  the  apparent  motion  of  the  sun,  then  the  direction  of 
magnetization  of  the  imaginary  shell  will  be  the  same  as  that  of  the  real  mag- 
netization of  the  earth*. 

The  total  intensity  of  magnetizing  force  in  a  closed  curve  passing  through 
and  embracing  the  closed  current  is  constant,  and  may  therefore  be  made  a 
measure  of  the  quantity  of  the  current.  As  this  intensity  is  independent  of  the 
form  of  the  closed  curve  and  depends  only  on  the  quantity  of  the  current  which 
passes  through  it,  we  may  consider  the  elementary  case  of  the  current  which 
Hows  through  the  elementary  area  dydz. 

Let  the  axis  of  x  point  towards  the  west,  z  towards  the  south,  and  y 
upwards.  Let  x,  y,  z  be  the  coordinates  of  a  point  in  the  middle  of  the  area 
dydz,  then  the  total  intensity  measured  round  the  four  sides  of  tlie  element  is 


(A*Si)* 

('■* 

t'  1')  *. 
dy     2j 

{*- 

■ff)^^. 

('- 

■tf)<'^- 

[dz 

-©''^*- 

Total  intensity  = 

The  quantity  of  electricity  conducted  through  the  elementary  area  dydz  is 
adydz,  and  therefore  if  we  define  the  measure  of  an  electric  current  to  be  the 
total  intensity  of  magnetizing  force  in  a  closed  curve  embracing  it,  we  shall  have 

^^^dl,_dy, 
'      dz       dy  ' 


h,. 


dy^      dai 
dx       dz 


_da,_d£, 
'      dy      dx 


-■  See  Experimental  Researches  (3265)  for  the  relations  between  the  electrical  and  magnetic  circuit, 
considered  as  mutiudly  embracing  curves. 


ON   Faraday's   lines   of   force.  195 

These  equations  enable  us  to  deduce  the  distribution  of  the  currents  of 
electricity  whenever  we  know  the  values  of  a,  y3,  y,  the  magnetic  intensities. 
If  a,  /3,  y  be  exact  differentials  of  a  function  of  x,  y,  z  with  respect  to  x,  y 
and  2  respectively,  then  the  values  of  a,,  h^,  c,  disappear;  and  we  know  that  the 
magnetism  is  not  produced  by  electric  currents  in  that  part  of  the  field  which 
we  are  investigating.  It  is  due  either  to  the  presence  of  permanent  magnetism 
within  the  field,  or  to  magnetising  forces  due  to  external  causes. 

We  may  observe  that  the  above  equations  give  by  differentiation 

^  +  ^'4.^^  =  0 

dx      dy      dz       * 

which  is  the  equation  of  continuity  for  closed  currents.  Our  investigations  are 
therefore  for  the  present  limited  to  closed  currents ;  and  we  know  little  of  the 
magnetic  effects  of  any  currents  which  are  not  closed. 

Before  entering  on  the  calculation  of  these  electric  and  magnetic  states  it 
may  be  advantageous  to  state  certain  general  theorems,  the  truth  of  which  may 
be  established  analytically. 

Theorem  I. 

The  equation 

d'V     d^V     d'V     ^         ^ 

d^-^W'^^'^  ^^^  ' 

(where  V  and  p  are  functions  of  x,  y,  z  never  infinite,  and  vanishing  for  all  points 
at  an  infinite  distance),  can  be  satisfied  by  one,  and  only  one,  value  of  V.  See 
Art.  (17)  above. 


Theorem  II. 

The   value  of   V    which   will   satisfy  the  above  conditions  is   found  by   inte- 
grating the  expression 

pdxdydz 


///, 


where  the  limits  of  x,  3/,  2  are  such   as   to  include  every  point  of  space   where  /> 
is  finite. 

25—2 


196  ON  Faraday's   lines   of  force. 

The  proofs  of  these  theorems  may  be   found  in  any  work   on   attractions  or 

electricity,  and  in  particular  in  Green's  Essay  on  the  Application  of  Mathematics 

to   Electricity.     See  Arts.    18,    19  of  this  paper.  See  also  Gauss,   on   Attractions^ 
translated  in  Taylor's  Scientijtc  Memoirs. 


Theorem  III. 
Let  U  and   V  be  two  functions  of  x,  y,  z,  then 

d'U     d'U     d'-U\  J.,    ,    , 

where   the   integrations  are   supposed   to   extend   over   all   the   space   in   which   U 
and   V  have  values  differing  from  0. — (Green,  p.  10.) 

This  theorem  shews  that  if  there  be  two  attracting  systems  the  actions 
between  them  are  equal  and  opposite.  And  by  making  U=  V  we  find  that 
the  potential  of  a  system  on  itself  is  proportional  to  the  integral  of  the  square 
of  the  resultant  attraction  through  all  space ;  a  result  deducible  from  Art.  (30), 
since  the  volume  of  each  cell  is  inversely  as  the  square  of  the  velocity  (Arts. 
12,  13),  and  therefore  the  number  of  cells  in  a  given  space  is  directly  as  the 
square  of  the  velocity. 

Theorem  IV. 

Let  a,  /8,  y,  p  be  quantities  finite  through  a  certain  space  and  vanishing 
in  the  space  beyond,  and  let  k  be  given  for  all  parts  of  space  as  a  continuous 
or  discontinuous  function  of  x,  y,  z,  then  the  equation  in  p 

has   one,   and   only   one   solution,    in    which    p   is   always   finite    and    vanishes   at 
an  infinite  distance. 

The  proof  of  this  theorem,  by  Prof  W.  Thomson,  may  be  found  in  the 
Cambridge  and  Dublin  Mathematical  Journal,  Jan.  1848. 


ON    FARADAY  S     LINES    OF     FORCE,  197 

If  a,  /3,  y  be  the  electro-motive  forces,  p  the  electric  tension,  and  Ic  the 
coefficient  of  resistance,  tlien  the  above  equation  is  identical  with  the  equation 
of  continuity 

da^  ,dh,dc, 

ax      dy      dz  r        ' 

and  the  theorem  shews  that  when  the  electro-motive  forces  and  the  rate  of 
production  of  electricity  at  every  part  of  space  are  given,  the  value  of  the 
electric  tension  is  determinate. 

Since  the  mathematical  laws  of  magnetism  are  identical  with  those  of  elec- 
tricity, as  far  as  we  now  consider  them,  we  may  regard  a,  /8,  y  as  magnetizing 
forces,  p  as  magnetic  tension,  and  p  as  real  magnetic  density,  k  being  the 
coefficient  of  resistance  to  magnetic  induction. 

The   proof  of  this   theorem  rests  on  the  determination  of  the  minimum  value 

where   V  is  got  from  the  equation 

d'V     d'V     d'V     , 

and  p  has  to  be  determined. 

The  meaning  of  this  integral  in  electrical  language  may  be  thus  brought 
out.  If  the  presence  of  the  media  in  which  k  has  various  values  did  not 
affect   the   distribution   of  forces,    then  the   '^quantity"    resolved   in   x   would   be 

simply  -7—  and  the   intensity   k  -^ .     But   the  actual  quantity  and   intensity  are 

J-  (a  —  j-j  and  a— ^,  and  the  parts  due  to  the  distribution  of  media  alone 
are  therefore 


1  /       dp\      dV       ,         dp     ,  dV 
T  {°'-~ji  — 7-  and  a  —  ~  —  k  -i-  . 
fc  \      ax)      dx  dx        dx 


Now  the  product  of  these  represents  the  work  done  on  account  of  this 
distribution  of  media,  the  distribution  of  sources  being  determined,  and  taking 
in   the   terms   in   y  and   z   we   get   the   expression    Q   for   the    total    work     done 


198  ON  Faraday's  lines  or  force. 

by   that   part   of  the   whole   effect   at   any  point  which  is  due  to  the  distribution 
of  conducting  media,  and  not  directly  to  the  presence  of  the  sources. 

This   quantity    Q   is   rendered   a   minimum   by   one   and  only  one  value  of  p, 
namely,  that  which  satisfies  the  original  equation. 

Theorem  V. 
If  a,  h,  c  be  three  functions  of  x,  y,  %  satisfying  the  equation 

da     db     ^  _r. 
dx     dy     dz~   ' 

it  is  always  possible  to  find  three  functions  a,  /3,  y  which  shall  satisfy  the  equa- 
tions 

dz      dy       ' 


i- 

da 

-h, 

da 

dfi 

-Tx'- 

=  c. 

Let  A  =  Icdy,  where  the  integration  is  to  be  performed  upon  c  considered 
as  a  function  of  y,  treating  x  and  z  as  constants.  Let  B='\adz,  C^\hdx, 
A'  =  \hdz,  R  =  \cdx,  C'  =  \ady,  integrated  in  the  same  way. 

Then 

will  satisfy  the  given  equations ;    for 

d§^_dy^fda^^^fdc^^__fdb^^_^fda  , 
dz      dy     J  dy  J  dz  Jdy  J  dy    ^' 

and  0=\--j-dx+\-f-  dx+  l-j-  dx; 

d3     dy      (da  ,        (da  ,        (da  , 

=  a. 


ON   Faraday's   lines   of   force.  199 

In  the  same  way  it  may  be  shewn  that  the  values  of  a,  ^,  y  satisfy 
the  other  given  equations.  The  function  i/;  may  be  considered  at  present  as 
perfectly  indeterminate. 

The  method  here  given  is  taken  from  Prof.  W.  Thomson's  memoir  on 
Magnetism  {Phil   Trans.   1851,  p.  283). 

As  we  cannot  perform  the  required  integrations  when  a,  h,  c  are  discon- 
tinuous functions  of  x,  y,  z,  the  following  method,  which  is  perfectly  general 
though  more  compUcated,  may  indicate  more  clearly  the  truth  of  the  proposition. 

Let  A,  B,  C  be  determined  from  the  equations 
d'A     d'A     d'A 

^  +  ^^  +  £^  +  6  =  0 
dor       dy^       dz'  ' 

d'Cd'Ccr-c^     ^ 

by  the   methods   of  Theorems   I.   and   II.,   so   that   A,    B,    C  are  never  infinite, 
and  vanish  when  x,  y,  or  z  is  infinite. 
Also  let 


then 


a  = 

dB 

-dz' 

dC 
-dy- 

d^ 
^dx' 

0- 

dC 
~  dx' 

dA 

"dz' 

dy 

7- 

dA 
-dy- 

dB 
~dx' 

drP 

^Tz' 

^ 

dB 

-dy' 

dC\      fd'A     d'A 
-  dz)      W  "^  clf  "^ 

d"-A 
dz\ 

d^/dA     dB     dC\,^ 
dx\dx      dy      dzj 


dx\dx      dy 

If  we  find  similar  equations  in  y  and  z,  and  differentiate  the  first  by  x, 
the  second  by  y,  and  the  third  by  z,  remembering  the  equation  between 
a,  b,  c,  we  shall  have 

/c?^       d^      dr\fdA      dB     cZC\ 

\dxr     dif'     dz^]\dx      dy       dz 


200  ON   Faraday's   lines  of   force. 

and   since   A,   B,    C    are    always    finite   and   vanish   at   an   ir finite   distance,    the 
only  solution  of  this  equation  is 

dA     dB     dC^^ 
dx      dy      dz       * 

and  we  have  finally 

d§  _dY_ 
dz      dy~   ' 

with  two  similar  equations,  shewing  that  a,  /9,  y  have  been  rightly  determined. 

The  function  i/»  is  to  be  determined  from  the  condition 

dx^  dy^  dz~  [dx"  '^dy'^  dz')  ^ ' 
if  the  left-hand  side  of  this  equation  be  always  zero,  xp  must  be  zero  also. 


Theorem  YI, 

Let   a,    h,    c   he  any   three  functions  of  x,  y,  z,  it  is  possible  to  find  three 
functions  a,  /8,  y  and  a  fourth   V,  so  that 

dx     dy      dz       ' 

and  =^_^     ^ 

dz      dy     dx  ' 

,_dy^dadV 
dx     dz      dy  ' 

dy     dx      dz 

Let 

da     dh     dc 

di  +  Ty  +  dz^-^^'P' 

and  let   V  be  found  from  the  equation 

d^V    d'V    d'V 


ON   fabaday's   lines   of   force.  201 

then 


a'^a- 

dV 
dx' 

h'  =  h- 

dV 

c=c- 

dV 

da' 
-dbc 

dh' 

dc     ^ 
dz 

satisfy  the  condition 


and  therefore  we  can  find  three  tunctions  A,  B,  C,  and  from  these  a,  ^,  y,  so  as 
to  satisfy  the  given  equations. 

Theorem  VIL 

The  integral  throughout  infinity 

Q  =  jjj  (a,a,  +  hfi,  +  c^y,)  dxdydz, 
where  a}>fi^,  a^{y^  are  any  functions  whatsoever,  is  capable  of  transformation  into 

Q=+  lll{^n>P^  -  (^o«2  +  A^2  +  roC,)}  dxdydz, 
in  which  the  quantities  are  found  from  the  equations 

dcL      dh,      d€< 
^■'dy^fz^'^P^-''^ 

ojSoyo^  axe  determined  from  ap^c^  by  the  last  theorem,  so  that 

^      dz       dy      dx  ' 
a}>/:^  are  found  from  cgSiyi  by  the  equations 

and  p  is  found  from  the  equation 

d'p     d'pd'p^^      ,     . 

vol.  l  26 


202  ON  Faraday's  lines  of  force. 

For,  if  we  put  a,  in  the  form 

dz       dy      dx  ' 

and   treat   h^   and  c,   similarly,   then   we   have   by  integration    by   parts    through 
infinity,  remembering  that  all  the  functions  vanish  at  the  limits, 

or  <?  =  +  ///{(47r  V)  -  (aA  +  A&.  +  y.c,)]  dxdydz, 
and  by  Theorem  III. 

Ill  Vp  dxdydz  =  lUppdxdydz, 
so  that  finally 

Q  =  lll{^7rpp  -  (a„a,  +  A^2  +  y«cj}  dxdydz. 

If  afi^c^  represent  the  components  of  magnetic  quantity,  and  a^iyi  those 
of  magnetic  intensity,  then  p  will  represent  the  real  magnetic  density,  and  p 
the  magnetic  potential  or  tension.  aJ)iCi  will  be  the  components  of  quantity 
of  electric  currents,  and  a^^.y^  will  be  three  functions  deduced  from  afi^c^, 
which  will  be  found  to  be  the  mathematical  expression  for  Faraday's  Electro- 
tonic  state. 

Let  us  now  consider  the  bearing  of  these  analytical  theorems  on  the 
theory  of  magnetism.  Whenever  we  deal  with  quantities  relating  to  magnetism, 
we  shall  distinguish  them  by  the  suffix  d).  Thus  aj^iC,  are  the  components 
resolved  in  the  directions  of  x,  y,  z  of  the  quantity  of  magnetic  induction  acting 
through  a  given  point,  and  aJS^yi  are  the  resolved  intensities  of  magnetization 
at  the  same  point,  or,  what  is  the  same  thing,  the  components  of  the  force 
which  would  be  exerted  on  a  unit  south  pole  of  a  magnet  placed  at  that 
point  without  disturbing  the  distribution  of  magnetism. 

The  electric  currents  are  found  from  the  magnetic  intensities  by  the  equations 

djB,      dy,  , 
dz       dy 

When  there  are  no  electric  currents,  then 

a^dx  +  P^dy  -f  y^dz  =  dp, , 


ON   Faraday's  lines   of   force.  203 

a  perfect  differential  of  a  function  of  x,  y,  z.     On   the    principle    of   analogy    we 
may  call  jo,  the  magnetic  tension. 

The  forces  which  act  on  a  mass  m  of  south  magnetism   at  any  point  are 

in   the   direction   of  the    axes,    and   therefore    the    whob    work   done   during   any 
displacement  of  a  magnetic  system  is  equal  to  the  decrement  of  the  integral 

Q  =  ll\p,p4xdydz 
throughout  the  system. 

Let  us  now  call  Q  the  total  potential  of  the  system  on  itself.  The  increase 
or  decrease  of  Q  will  measure  the  work  lost  or  gained  by  any  displacement 
of  any  part  of  the  system,  and  will  therefore  enable  us  to  determine  the 
forces  acting  on  that  part  of  the  system. 

By  Theorem  III.  Q  may  be  put  under  the  form 


Q  =  +  ^  j    I  (ctio,  +  hSi  +  c,y,)  dxdydz 


in  which  a^iji  are  the  differential  coefficients  of  p^  with  respect  to  x,  y,  z 
respectively. 

If  we  now  assume  that  this  expression  for  Q  is  true  whatever  be  the 
values  of  Oj,  )8„  yi,  we  pass  from  the  consideration  of  the  magnetism  of  permanent 
magnets  to  that  of  the  magnetic  effects  of  electric  currents,  and  we  have  then 
by  Theorem  VII. 

So  that  in  the  case  of  electric  currents,  the  components  of  the  currents  have 
to  be  multiplied  by  the  functions  a„,  ySj,  yo  respectively,  and  the  summations  of 
all  such  products  throughout  the  system  gives  us  the  part  of  Q  due  to  those 
currents. 

We  have  now  obtained  in  the  functions  a,,,  Aj  yo  the  means  of  avoiding 
the  consideration  of  the  quantity  of  magnetic  induction  which  passes  through 
the  circuit.  Instead  of  this  artificial  method  we  have  the  natural  one  of  con- 
sidering the  current  with  reference  to  quantities  existing  in  the  same  space 
with  the  current  itself.  To  these  I  give  the  name  of  Electro-tonic  functions,  or 
components  of  the  Electro-tonic  intensity. 

2G— 2 


204  ON  Faraday's  lines  of  force. 

Let  us  now  consider  the  conditions  of  the  conduction  of  the  electric 
currents  within  the  medium  during  changes  in  the  electro-tonic  state.  The 
method  which  we  shall  adopt  is  an  appHcation  of  that  given  by  Helmholtz  in 
his  memoir  on  the  Conservation  of  Force*. 

Let  there  be  some  external  source  of  electric  currents  which  would  generate 
in  the  conducting  mass  currents  whose  quantity  is  measured  by  a^,  h^,  c,  and 
their  intensity  by  cu,,  /Sa,  y^. 

Then  the  amount  of  work  due  to  this  cause  in  the  time  dt  is 
dt  lll{a^(h  +  hS^  +  c^y^  dxdydz 
in  the  form  of  resistance  overcome,  and 

^  ^  J  j  J  (^2^0  4-  6  A  +  c,yo)  dxdydz 

in  the  form  of  work  done  mechanically  by  the  electro-magnetic  action  of  these 
currents.  If  there  be  no  external  cause  producing  currents,  then  the  quantity 
representing  the  whole  work  done  by  the  external  cause  must  vanish,  and  we 
have 

dt  \\  \(a,a^  +  hS,  +  c.y,)  dxdydz  +  4^  ^  I  I  I  («**o  +  ^So  +  c^Jo)  dxdydz, 

where  the  integrals  are  taken  through  any  arbitrary  space.  We  must  therefore 
have 

for  every  point  of  space ;  and  it  must  be  remembered  that  the  variation  of 
Q  is  supposed  due  to  variations  of  a^,  ySo,  y^,  and  not  of  a^,  \,  c^.  We  must 
therefore  treat   a^,  63,  c^  as  constants,  and  the  equation  becomes 

In  order  that  this  equation  may  be  independent  of  the  values  of  a^,  b^,  Cj, 
each  of  these  coefficients  must  =  0 ;  and  therefore  we  have  the  following 
expressions  for  the  electro-motive  forces  due  to  the  action  of  magnets  and 
currents  at  a  distance  in  terms  of  the  electro-tonic  functions, 

°^~      ATrdt'  ^^~      Andt'         '^'~      An  dt  ' 

*  Translated  in  Taylor's  N'ew  Scientific  Memoirs,  Part  11. 


ON   Faraday's  lines   of   force.  205 

It   appears   from   experiment  that   the   expression   -jj   refers    to    the    change 

of    electro-tonic    state    of    a    given    particle    of   the    conductor,    whether   due   to 
change  in  the  electro-tonic  functions  themselves  or  to  the  motion  of  the  particle. 

If  Oo  be  expressed  as  a  function  of  x,  y,  z  and  t,  and  \£  x,  y,  z  be  the 
co-ordinates  of  a  moving  particle,  then  the  electro-motive  force  measured  in  the 
direction  of  a;  is 


_  _  Jl  (^'  dx     da^dy     da,dz     doA 
°^~      477  \dx  dt      dy  dt      dz  dt      dtj 


The  expressions  for  the  electro-motive  forces  in  y  and  z  are  similar.  The 
distribution  of  currents  due  to  these  forces  depends  on  the  form  and  arrange- 
ment of  the  conducting  media  and  on  the  resultant  electric  tension  at  any 
point. 

The  discussion  of  these  functions  would  involve  us  in  mathematical  formulae, 
of  which  this  paper  is  already  too  full.  It  is  only  on  account  of  their  physical 
importance  as  the  mathematical  expression  of  one  of  Faraday's  conjectures  that  I 
have  been  induced  to  exhibit  them  at  all  in  their  present  form.  By  a  more 
patient  consideration  of  their  relations,  and  with  the  help  of  those  who  are 
engaged  in  physical  inquiries  both  in  this  subject  and  in  others  not  obviously 
connected  with  it,  I  hope  to  exhibit  the  theory  of  the  electro-tonic  state  in  a 
form  in  which  all  its  relations  may  be  distinctly  conceived  without  reference  to 
analytical  calculations. 


Summary  of  the  Theory  of  the  Electro-tonic  State. 

We  may  conceive  of  the  electro-tonic  state  at  any  point  of  space  as  a 
quantity  determinate  in  magnitude  and  direction,  and  we  may  represent  the 
electro-tonic  condition  of  a  portion  of  space  by  any  mechanical  system  which 
has  at  every  point  some  quantity,  which  may  be  a  velocity,  a  displacement,  or 
a  force,  whose  direction  and  magnitude  correspond  to  those  of  the  supposed 
electro-tonic  state.  This  representation  involves  no  physical  theory,  it  is  only 
a  kind  of  artificial  notation.  In  analytical  investigations  we  make  use  of  the 
three  components  of  the  electro-tonic  state,  and  call  them  electro-tonic  functions. 
We   take    the   resolved    part   of    the  electro-tonic   intensity   at    every  point   of  a 


206  ON  Faraday's  lines  of   force. 

closed  curve,  and  find  by  integration  what  we  may  caU  the  entire  electro-tonic 
intensity  round  the  curve. 

Prop.  I.  If  on  any  surface  a  closed  curve  be  drawn,  and  if  the  surface 
within  it  he  divided  into  small  areas,  then  the  entire  intensity  round  the  closed 
curve  is  equal  to  the  sum  of  the  intensities  round  each  of  the  small  areas,  all 
estimated   in   the   same   direction. 

For,  in  going  round  the  small  areas,  every  boundary  line  between  two  of 
them  is  passed  along  twice  in  opposite  directions,  and  the  intensity  gained  in 
the  one  case  is  lost  in  the  other.  Every  eflfect  of  passing  along  the  interior 
divisions  is  therefore  neutraUzed,  and  the  whole  efiect  is  that  due  to  the 
exterior  closed   curve. 

Law  I.  The  entire  dectro-tonic  intensity  round  the  boundary  of  an  element  of 
surface  measures  the  quantity  of  magnetic  induction  which  passes  through  that 
surface,  or,  in  other  words,  the  number  of  lines  of  magnetic  force  which  pass 
through   that   surface. 

By  Prop.  I.  it  appears  that  what  is  true  of  elementary  surfaces  is  true  also 
of  surfaces  of  finite  magnitude,  and  therefore  any  two  surfaces  which  are 
bounded  by  the  same  closed  curve  will  have  the  same  quantity  of  magnetic 
induction  through  them. 

Law  II.  The  magnetic  intensity  at  any  point  is  connected  with  the  quantity 
of  magnetic  induction  by  a  set  of  linear  equations,  called  the  equations  of  con- 
duction*. 

Law  III.  The  entire  magnetic  intensity  round  the  boundary  of  any  surface 
measures  the  quantity  of  electric  current  which  passes  through  that  surface. 

Law  IV.  The  quantity  and  intensity  of  electric  currents  are  connected  by  a 
system  of  equations  of  conduction. 

By  these  four  laws  the  magnetic  and  electric  quantity  and  intensity  may  be 
deduced  from  the  values  of  the  electro-tonic  functions.  I  have  not  discussed 
the  values  of  the  units,  as  that  will  be  better  done  with  reference  to  actual 
experiments.  We  come  next  to  the  attraction  of  conductors  of  currents,  and  to 
the  induction  of  currents  within  conductors. 

*  See  Art.  (28). 


ON     FARADAY  S    LINES    OF    FORCE.  207 

Law  v.  The  total  electro-magnetic  potential  of  a  closed  current  is  measxired 
by  the  product  of  the  quantity  of  the  current  multiplied  by  the  entire  electro-tonic 
intensity  estimated  in  t/ie  same  direction  round  the  circuit. 

Any  displacement  of  the  conductors  which  would  cause  an  increase  in  the 
potential  will  be  assisted  by  a  force  measured  by  the  rate  of  increase  of  the 
potential,  so  that  the  mechanical  work  done  during  the  displacement  will  be 
measured  by  the  increase  of  potential. 

Although  in  certain  cases  a  displacement  in  direction  or  alteration  of  inten- 
sity of  the  current  might  increase  the  potential,  such  an  alteration  would  not 
itself  produce  work,  and  there  will  be  no  tendency  towards  this  displacement, 
for  alterations  in  the  current  are  due  to  electro-motive  force,  not  to  electro- 
magnetic attractions,  which  can  only  act  on  the  conductor. 

Law  VI.  The  electro-motive  force  on  any  element  of  a  conductor  is  measured 
by  the  instantaneous  rate  of  change  of  the  electro-tonic  intensity  on  that  element, 
whether  in  magnitude  or  direction. 

The  electro-motive  force  in  a  closed  conductor  is  measured  by  the  rate  of 
change  of  the  entire  electro-tonic  intensity  round  the  circuit  referred  to  unit 
of  time.  It  is  independent  of  the  nature  of  the  conductor,  though  the  current 
produced  varies  inversely  as  the  resistance ;  and  it  is  the  same  in  whatever 
way  the  change  of  electro-tonic  intensity  has  been  produced,  whether  by  motion 
of  the  conductor  or  by  alterations  in  the  external  circumstances. 

In  these  six  laws  I  have  endeavoured  to  express  the  idea  which  I  believe  to 
be  the  mathematical  foundation  of  the  modes  of  thought  indicated  in  the  Ex- 
perimental Researches.  I  do  not  think  that  it  contains  even  the  shadow  of  a 
true  physical  theory;  in  fact,  its  chief  merit  as  a  temporary  instrument  of 
research  is  that  it  does  not,  even  in  appearance,  account  for  anything. 

There  exists  however  a  professedly  physical  theory  of  electro-dynamics,  which 
is  so  elegant,  so  mathematical,  and  so  entirely  different  from  anything  in  this 
paper,  that  I  must  state  its  axioms,  at  the  risk  of  repeating  what  ought  to 
be  well  known.  It  is  contained  in  M.  W.  Weber's  Electro-dynamic  Measure- 
ments, and  may  be  found  in  the  Transactions  of  the  Leibnitz  Society,  and  of  the 
Royal  Society  of  Sciences  of  Saxony*.     The  assumptions  are, 

*  When  this  was  written,  I  was  not  aware  that  part  of  M.  Weber's  Memoir  is  translated  in 
Taylor's  Scientific  Memoirs,  VoL  v.  Art.  xiv.  The  value  of  his  researches,  both  experimental  and 
theoretical,  renders  the  study  of  his  theory  necessary  to  every  electrician. 


208  ON   Faraday's   lines   of  force. 

(1)  That  two  particles  of  electricity  when  in  motion  do  not  repel  each  other 
with  the  same  force  as  when  at  rest,  but  that  the  force  is  altered  by  a  quantity 
depending  on  the  relative  motion  of  the  two  particles,  so  that  the  expression  for 
the   repulsion   at   distance   r   is 


eeV,        dr 


(2)  That  when  electricity  is  moving  in  a  conductor,  the  velocity  of  the 
positive  fluid  relatively  to  the  matter  of  the  conductor  is  equal  and  opposite  to 
that  of  the  negative  fluid. 

(3)  The  total  action  of  one  conducting  element  on  another  is  the  resultant 
of  the  mutual  actions  of  the  masses  of  electricity  of  both  kinds  which  are 
in   each. 

(4)  The  electro-motive  force  at  any  point  is  the  difference  of  the  forces 
acting   on   the   positive   and   negative   fluids. 

From  these  axioms  are  deducible  Ampere's  laws  of  the  attraction  of 
conductors,  and  those  of  Neumann  and  others,  for  the  induction  of  currents. 
Here  then  is  a  really  physical  theory,  satisfying  the  required  conditions  better 
perhaps  than  any  yet  invented,  and  put  forth  by  a  philosopher  whose  experi- 
mental researches  form  an  ample  foundation  for  his  mathematical  investigations. 
What  is  the  use  then  of  imagining  an  electro-tonic  state  of  which  we  have 
no  distinctly  physical  conception,  instead  of  a  formula  of  attraction  which  we 
can  readily  understand  ?  I  would  answer,  that  it  is  a  good  thing  to  have 
two  ways  of  looking  at  a  subject,  and  to  admit  that  there  are  two  ways  of 
looking  at  it.  Besides,  I  do  not  think  that  we  have  any  right  at  present  to 
understand  the  action  of  electricity,  and  I  hold  that  the  chief  merit  of  a 
temporary  theory  is,  that  it  shall  guide  experiment,  without  impeding  the 
progress  of  the  true  theory  when  it  appears.  There  are  also  objections  to 
making  any  ultimate  forces  in  nature  depend  on  the  velocity  of  the  bodies 
between  which  they  act.  If  the  forces  in  nature  are  to  be  reduced  to  forces 
acting  between  particles,  the  principle  of  the  Conservation  of  Force  requires 
that  these  forces  should  be  in  the  line  joining  the  particles  and  functions  of 
the  distance  only.  The  experiments  of  M.  Weber  on  the  reverse  polarity  of 
diaraagnetics,  which  have  been  recently  repeated  by  Professor  Tyndall,  establish 
a  fact  which  is  equally  a  consequence  of  M.  Weber's  theory  of  electricity  and 
of  the   theory  of  lines   of  fcH-ce. 


ON    FARADAY  S    LINES    OF    FORCE.  209 

With  respect  to  the  history  of  the  present  theory,  I  may  state  that  the 
recognition  of  certain  mathematical  functions  as  expressing  the  "electro-tonic 
state "  of  Faraday,  and  the  use  of  them  in  determining  electro-dynamic 
potentials  and  electro-motive  forces  is,  as  far  as  I  am  aware,  original ;  but  the 
distinct  conception  of  the  possibility  of  the  mathematical  expressions  arose  in 
my  mind  from  the  perusal  of  Prof  W.  Thomson's  papers  "On  a  Mechanical 
Representation  of  Electric,  Magnetic  and  Galvanic  Forces,"  Cambridge  and 
Dublin  Mathematical  Journal,  January,  1847,  and  his  "Mathematical  Theory  of 
Magnetism,"  Philosophical  Transactions,  Part  I.  1851,  Art.  78,  &c.  As  an 
instance  of  the  help  which  may  be  derived  from  other  physical  investigations, 
I  may  state  that  after  I  had  investigated  the  Theorems  of  this  paper 
Professor  Stokes  pointed  out  to  me  the  use  which  he  had  made  of  similar 
expressions  in  his  "Dynamical  Theory  of  Diffraction,"  Section  1,  Camhndge 
Transactions,  Vol.  ix.  Part  1.  Whether  the  theory  of  these  functions,  consi- 
dered with  reference  to  electricity,  may  lead  to  new  mathematical  ideas  to  be 
employed  in  physical  research,  remains  to  be  seen.  I  propose  in  the  rest  of 
this  paper  to  discuss  a  few  electrical  and  magnetic  problems  with  reference  to 
spheres.  These  are  intended  merely  as  concrete  examples  of  the  methods  of 
which  the  theory  has  been  given ;  I  reserve  the  detailed  investigation  of  cases 
chosen  with  special  reference  to  experiment  till  I  have  the  means  of  testing 
their  results. 


Examples. 
I.     Theory  of  Electrical  Images. 

The  method  of  Electrical  Images,  due  to  Prof  W.  Thomson"",  by  whicli 
the  theory  of  spherical  conductors  has  been  reduced  to  great  geometrical  sim- 
plicity, becomes  even  more  simple  when  we  see  its  connexion  with  the  methods 
of  this  paper.  We  have  seen  that  the  pressure  at  any  point  in  a  uniform 
medium,    due    to   a   spherical   shell    (radius  =  a)    giving   out    fluid   at    the   rate   of 

a" 
AnPa^    units    in    unit   of    time,    is   ^P—    outside    the   shell,    and   kPa    inside   it, 

r 

where  r  is  the  distance  of  the  point  from  the  centre  of  the  shell. 

*  See   a   series   of  papers    "On   the   Mathematical   Theory  of  Electricity,"  in   the   Cambridge  and 
Dublin  Math.  Jour.,  beginning  March,  1848. 

VOL    L  27 


210  ON   Faraday's  lines   of  force. 

If  there  be  two  shells,  one  giving  out  fluid  at  a  rate  inPa\  and  the 
other  absorbing  at  the  rate  of  iirFa\  then  the  expression  for  the  pressure  will 
be,  outside  the  shells, 

J^  r  r 

where  r  and  /  are  the  distances   from  the  centres   of  the  two  shells.     Equating 
this  expression  to  zero  we  have,  as  the  surface  of  no  pressure,  that  for  which 


/  _  Fa'' 
r  ~  Pa' 


Now  the   surface,   for  which  the  distances  to  two  fixed  points  hav^e  a  given 

ratio,  is   a   sphere   of  which  the  centre  0   is  in   the  line  joining  the   centres   of 
the  shells  CC   produced,  so  that 

and  its  radius  ^  ^ 


Pa'lt-F^' 

If  at  the  centre  of  this  sphere  we  place  another  source  of  the  fluid,  then 
the  pressure  due  to  this  source  must  be  added  to  that  due  to  the  other  two; 
and  since  this  additional  pressure  depends  only  on  the  distance  from  the  centre, 
it  will  be  constant  at  the  surface  of  the  sphere,  where  the  pressure  due  to 
the  two  other  sources  is  zero. 

We  have  now  the  means  of  arranging  a  system  of  sources  within  a  given 
sphere,  so  that  when  combined  with  a  given  system  of  sources  outside  the 
sphere,  they  shall  produce  a  given  constant  pressure  at  the  surface  of  the  sphere. 

Let  a  be  the  radius  of  the  sphere,  and  p  the  given  pressure,  and  let  the 
given  sources  be  at  distances  6„  h„  &c.  from  the  centre,  and  let  their  rates  of 
production  be  4.TrP„.  47rP„  &c. 

Then  if  at  distances  ^ ,  ?- ,  &c.  (measured  in  the  same  direction  as  h„  \,  &c. 
from  the  centre)  we  place  negative  sources  whose  rates  are 

-47rP,?,     -477P,^,  &c., 

0,  Oj 


ON   Faraday's   lines   of   force.  211 

the  pressure  at  the  surface  r  =  a  will  be  reduced  to  zero.  Now  placing  a  source 
477-^  at  the  centre,  the  pressure  at  the  surface  will  be  uniform  and  equal  to  />. 

The  whole  amount  of  fluid  emitted  by  the  surface  r  =  a  may  be  found  by 
adding  the  rates  of  production  of  the  sources  within  it.     The  result  is 

To  apply  this  result  to  the  case  of  a  conducting  sphere,  let  us  suppose 
the  external  sources  inP^,  AnP^  to  be  small  electrified  bodies,  containing  e„  e, 
of  positive  electricity.  Let  us  also  suppose  that  the  whole  charge  of  the  con- 
ducting sphere  is  =E  previous  to  the  action  of  the  external  points.  Then  all 
that  is  required  for  the  complete  solution  of  the  problem  is,  that  the  surface 
of  the  sphere  shall  be  a  surface  of  equal  potential,  and  that  the  total  charge 
of  the  surface  shall  be  E. 

If  by  any  distribution  of  imaginary  sources  within  the  spherical  surface  we 
can  effect  this,  the  value  of  the  corresponding  potential  outside  the  sphere  is 
the  true  and  only  one.  The  potential  inside  the  sphere  must  really  be  constant 
and  equal  to  that  at  the  surface. 

We  must  therefore  find  the  images  of  the  external  electrified  points,  that 
is,   for  every   point   at  distance    b   from   the   centre  we  must  find  a  point  on  the 

same  radius  at  a  distance  j- ,  and  at  that  point  we  must  place  a  quantity 
=  —  e  ,    of  imaginary  electricity. 

At  the  centre  we  must  put  a  quantity  E'  such  that 
K  =  E  +  e,^  +  e,^-  +  kc.; 

then  if  i^  be  the  distance  from  the  centre,  r„  r^,  &c.  the  distances  from  the 
electrified  points,  and  r\,  r\,  &c.  the  distances  from  their  images  at  any  point 
outside  the  sphere,  the  potential  at  that  point  will  be 

E     e,  ( a      \      ci\      e,  /a      b,      a\  ,  . 


212  ON  Faraday's  lines  of  force. 

This   is   the   value   of  the   potential   outside   the  sphere.     At  the  surface  we 


have 


K  =  a  and  —  =  -7- ,        —  =  -7- ,   ac. 


so  that  at  the  surface 


and  this  must  also  be  the  value  oi  p  for  any  point  within  the  sphere. 

For  the  application  of  the  principle  of  electrical  images  the  reader  is  referred 
to   Prof   Thomson's  papers  in  the   Cambridge  and  Dublin  Mathematical  Journal. 

The   only   case  which  we   shall  consider  is   that  in   which  A  =  /,  and   b^  is  infi- 
nitely distant  along  the  axis  of  x,  and  j&=0. 

The  value  p  outside  the  sphere  becomes  then 

and  inside  ^  =  0. 


II.     On  the  effect  of  a  paramagnetic  or  diam/xgnetic  sphere  in  a  uniform  field  oj 

magnetic  force'^. 

The  expression  for  the  potential  of  a  small  magnet  placed  at  the  origin  of 
co-ordinates  in  the  direction  of  the  axis  of  x  is 


dx  \rj~ 


i:^i'-]=-lm^ 


The  eflPect  of  the  sphere  in  disturbing  the  lines  of  force  may  be  supposed 
as  a  first  hypothesis  to  be  similar  to  that  of  a  small  magnet  at  the  origin, 
whose  strength  is  to  be  determined.     (We  shall  find  this  to  be  accurately  true.) 

*  See  Prof.  Thomson,  on  the  Theory  of  Magnetic  Induction,  PhiL  Mag.  March,  1851.     The  induc- 
tive capadiy  of  the  sphere,  according  to  that  paper,  is  the  ratio  of  the  qv/iTiiUy  of  magnetic  induction 

(not  the  intensity)  within  the  sphere  to  that  without     It  is  therefore  equal  to  j^T  =  2k    k'  ^^^^^' 
ing  to  our  notation. 


ON  Faraday's  lines  of  force.  213 

Let  the  value  of  the  potential  undisturbed  by  the  presence  of  the  sphere  be 

'p  =  Ix. 
Let  the  sphere  produce  an  additional  potential,  which  for  external  points  is 

,      .  a' 

and  let  the  potential  within  the  sphere  be 

Pi  =  Bx. 

Let  k'  be  the  coefficient  of  resistance  outside,  and  k  inside  the  bphere,  then 
the  conditions  to  be  fulfilled  are,  that  the  interior  and  exterior  potentials  should 
coincide  at  the  surface,  and  that  the  induction  through  the  surface  should  be  the 
same  whether  deduced  from  the  external  or  the  internal  potential.  Putting 
a;  =  rcos^,  we  have  for  the  external  potential 

P  =  //r  +  ^^')cos^, 

and  for  the  internal 

p^  =  Brco%dy 

and  these  must  be  identical  when  r  =  a,  or 

I+A  =  B. 

The  induction  through  the  surface  in  the  external  medium  is 

and  that  through  the  interior  surface  is 

and  .•.  i(7-2^)  =  i£. 


These  equations  give 


A  =  ^^f^J,  B=     ^^ 


2k  +  k'    '  ik  +  k' 


The    effect    outside   the   sphere  is   equal   to   that   of   a   little   magnet   whose 
length  is  I  and  moment  ml,  provided 


214 


ON  Faraday's  lines  of  force. 


Suppose  this  uniform  field  to  be  that  due  to  terrestrial  magnetism,  then, 
if  k  is  less  than  k'  as  in  paramagnetic  bodies,  the  marked  end  of  the  equi- 
valent magnet  will  be  turned  to  the  north.  If  A;  is  greater  than  F  as  in 
diamagnetic  bodies,  the  unmarked  end  of  the  equivalent  magnet  would  be  turned 
to  the  north. 


III.     Magnetic  Jield  of  variable  Intensity. 


Now  suppose  the  intensity  in  the  undisturbed  magnetic  field  to  vary  in 
magnitude  and  direction  from  one  point  to  another,  and  that  its  components 
in  X,  y,  z  are  represented  by  a,  /8,  y,  then,  if  as  a  first  approximation  we  re- 
gard the  intensity  within  the  sphere  as  sensibly  equal  to  that  at  the  centre, 
the  change  of  potential  outside  the  sphere  arising  from  the  presence  of  the 
sphere,  disturbing  the  lines  of  force,  will  be  the  same  as  that  due  to  three 
small  magnets  at  the  centre,  with  their  axes  parallel  to  x,  y,  and  z,  and  their 
moments  equal  to 

k-k'     3  k-k'     5^  k-k' 

2kTk'^^'       2FFF^^'        2FfF"^- 

The  actual  distribution  of  potential  within  and  without  the  sphere  may  be 
conceived  as  the  result  of  a  distribution  of  imaginary  magnetic  matter  on  the 
surface  of  the  sphere  ;  but  since  the  external  effect  of  this  superficial  magnetism 
is  exactly  the  same  as  that  of  the  three  small  magnets  at  the  centre,  the 
mechanical  effect  of  external  attractions  will  be  the  same  as  if  the  three  ma^ets 
really  existed. 

Now  let  three  small  magnets  whose  lengths  are  l^,  k,  k,  and  strengths 
m„  m^,  m„  exist  at  the  point  x,  y,  z  with  their  axes  parallel  to  the  axes  of 
then  resolving  the  forces  on  the  three  magnets  in  the  direction  of  X,  we 


X,  y,  z 
have 


X  =  'm^ 


da  Zi 


•a  + 


da  l^ 
dx  2 


Y  +'in.-{ 


a  + 


a  + 


da  I, 
dy  2 

da  It 

dy2\ 


■+«i. 


da  /g" 


a  + 


da  Zj 
dz  2. 


J  da  T  da  ,  da 


ON  Faraday's  lines   of  force.  215 


Substituting  the  values  of  the  moments  of  the  imaginary  magnets 
J    da     ^(IB        dy\       k-k'   a'    d   ,  ,     r>^  ,     2\ 


2k +  k' 

The  force  impelling  the  sphere  in  the  direction  of  x  is  therefore  dependent 
on  the  variation  of  the  square  of  the  intensity  or  (a'  +  ^  +  y),  as  we  move  along 
the  direction  of  x,  and  the  same  is  true  for  y  and  z,  so  that  the  law  is,  that 
the  force  acting  on  diamagnetic  spheres  is  from  places  of  greater  to  places  of 
less  intensity  of  magnetic  force,  and  that  in  similar  distributions  of  magnetic 
force  it  varies  as  the  mass  of  the  sphere  and  the  square  of  the  intensity. 

It  is  easy  by  means  of  Laplace's  CoeflBcients  to  extend  the  approximation 
to  the  value  of  the  potential  as  far  as  we  please,  and  to  calculate  the  attrac- 
tion. For  instance,  if  a  north  or  south  magnetic  pole  whose  strength  is  M,  be 
placed  at  a  distance  b  from  a  diamagnetic  sphere,  radius  a,  the  repulsion  will  be 

When  r  is  small,  the  first  term  gives  a  sufficient  approximation.     The  repul- 

0 

sion  is  then  as  the  square  of  the  strength  of  the  pole,  and  the  mass  of  the 
sphere  directly  and  the  fifth  power  of  the  distance  inversely,  considering  the 
pole  as  a  point. 


IV.     Tivo  Spheres  in  uniform  jield. 

Let  two  spheres  of  radius  a  be  connected  together  so  that  their  centres  are 
kept  at  a  distance  h,  and  let  them  be  suspended  in  a  uniform  magnetic  field, 
then,  although  each  sphere  by  itself  would  have  been  in  equilibrium  at  any  part 
of  the  field,  the  disturbance  of  the  field  will  produce  forces  tending  to  make  the 
balls  set  in  a  particular  direction. 

Let  the  centre  of  one  of  the  spheres  be  taken  as  origin,  then  the  undis- 
turbed potential  is 

p  =  Ir  cos  dy  ■ 

and  the  potential  due  to  the  sphere  is 

^  k  —  k'   a?        a 


216 


ON  Faraday's  lines  of  force. 


The  whole  potential  is  therefore  equal  to 

l(r  + 


'2jc  +  k' 


dp 
dr 


,^003  0=  p.. 


dp 
dr 


\ldp 


Idp 
rdS 

1 


dp\ 


|=«- 


T^^m'Bdi 


^'{i+^'^*(i-3-«''')+i5r^'(i+3-''')} 


This  is  the  value  of  the  square  of  the  intensity  at  any  point.  The  moment 
of  the  couple  tending  to  turn  the  combination  of  balls  in  the  direction  of  the 
original  force 

L  =  l^a^i7;fn?<n  when  r  =  h, 


dd    \2k  +  k' 


L^^P 


k-k' 


2k-\-k' 


k  —  k'  a\   .    ^^ 


This  expression,  which  must  be  positive,  since  h  is  greater  than  a,  gives  the 
moment  of  a  force  tending  to  turn  the  line  joining  the  centres  of  the  spheres 
towards  the  original  lines  of  force. 

Whether  the  spheres  are  magnetic  or  diamagnetic  they  tend  to  set  in  the 
axial  direction,  and  that  without  distinction  of  north  and  south.  If,  however, 
one  sphere  be  magnetic  and  the  other  diamagnetic,  the  line  of  centres  will  set 
equatoreally.  The  magnitude  of  the  force  depends  on  the  square  of  (k  —  k'),  and 
is  therefore  quite  insensible  except  in  iron*. 


V.     Two  Spheres  between  the  poles  of  a  Magnet. 

Let  us  next  take  the  case  of  the  same  balls  placed  not  in  a  uniform  field 
but  between  a  north  and  a  south  pole,  ±M,  distant  2c  from  each  other  in  the 
direction  of  x. 


*  See  Prof.  Thomson  in  Phil.  Mag.  March,  1851. 


ON   Faraday's  lines  of  force.  217 

The  expression  for  the  potential,  the  middle  of  the  line  joining  the  poles 
being  the  origin,  is 

p=m(,  '  —, ' )■ 

Wc*  +  i^-2crcos0     Vc'  +  ?-'  +  2crcos^/ 
From  this  we  find  as  the  value  of  P, 

P  =  i^7l_3!:+9^,cos-<^): 

c*     \  C^  &  ] 

.'.  I~=  -  18  ^^V sin  2^. 

and  the  moment  to  turn  a  pair  of  spheres  (radius  a,  distance  2h)  in  the 
direction  in  which  0  is  increased  is 

-^'wvk'-^''''^^' 

This  force,  which  tends  to  turn  the  line  of  centres  equatoreally  for  diamagnetic 
and  axially  for  magnetic  spheres,  varies  directly  as  the  square  of  the  strength  of 
the  magnet,  the  cube  of  the  radius  of  the  spheres  and  the  square  of  the  dis- 
tance of  their  centres,  and  inversely  as  the  sixth  power  of  the  distance  of  the 
poles  of  the  magnet,  considered  as  points.  As  long  as  these  poles  are  near  each 
other  this  action  of  the  poles  will  be  much  stronger  than  the  mutual  action  of 
the  spheres,  so  that  as  a  general  rule  we  may  say  that  elongated  bodies  set 
axially  or  equatoreally  between  the  poles  of  a  magnet  according  as  they  are  mag- 
netic or  diamagnetic.  If,  instead  of  being  placed  between  two  poles  very  near 
to  each  other,  they  had  been  placed  in  a  uniform  field  such  as  that  of  terrestrial 
magnetism  or  that  produced  by  a  spherical  electro-magnet  (see  Ex.  viii.),  an 
elongated  body  would  set  axially  whether  magnetic  or  diamagnetic. 

In  all  these  cases  the  phenomena  depend  on  k  —  k',  so  that  the  sphere  con- 
ducts itself  magnetically  or  diamagnetically  according  as  it  is  more  or  less 
magnetic,  or  less  or  more  diamagnetic  than  the  medium  in  which  it  is  placed. 


VI.     On  the  Magnetic  Phenomena  of  a  Sphere  cut  from  a  substance  whose 
coefficient  of  resistance  is  diffierent  in  different  directions. 

Let   the  axes  of  magnetic  resistance  be  parallel  throughout  the  sphere,  and 

let   them   be  taken  for  the  axes  of  x,  y,  z.     Let   K,  k„  k„  be  the  coefficients  of 

resistance   in  these  three  directions,  and  let  k'  be  that  of  the  external  medium, 

VOL.  I.  28 


218  ON    FARADAY  S    LINES    OF    FORCE. 

and  a  the  radius  of  the  sphere.  Let  /  be  the  undisturbed  magnetic  intensity 
of  the  field  into  which  the  sphere  is  introduced,  and  let  its  direction- cosines 
be  I,  m,  n. 

Let  us  now  take  the  case  of  a  homogeneous  sphere  whose  coefficient  is  ^, 
placed  in  a  uniform  magnetic  field  whose  intensity  is  II  in  the  direction  of  x. 
The  resultant  potential  outside  the  sphere  would  be 

and  for  internal  points 

So  that  in  the  interior  of  the  sphere  the  magnetization  is  entirely  in  the  direc- 
tion of  X.  It  is  therefore  quite  independent  of  the  coefficients  of  resistance  in 
the  directions  of  x  and  y,  which  may  be  changed  from  X\  into  k^  and  ^3  with- 
out disturbing  this  distribution  of  magnetism.  We  may  therefore  treat  the  sphere 
as  homogeneous  for  each  of  the  three  components  of  /,  but  we  must  use  a 
different  coefficient  for  each.     We  find  for  external  points 

and  for  internal  points 

The  external  effect  is  the  same  as  that  which  would  have  been  produced 
if  the  small  magnet  whose  moments  are 

te§'^"''    ^™^"''    te^'"-^"*' 

had  been  placed  at  the  origin  with  their  directions  coinciding  with  the  axes  of 
Xy  y,  z.  The  effect  of  the  original  force  /  in  turning  the  sphere  about  the  axis 
of  x  may  be  found  by  taking  the  moments  of  the  components  of  that  force 
on  these  equivalent  magnets.  The  moment  of  the  force  in  the  direction  of  y 
acting  on  the  third  magnet  is 

and  that  of  the  force  in  z  on  the  second  magnet  is 

2k^-\-k 


ON    FARADAY  S     LINES    OF    FORCE.  219 

The  whole  couple  about  the  axis  of  a;  is  therefore 

tending  to  turn  the  sphere  round  from  the  axis  of  y  towards  that  of  z.  Sup- 
pose the  sphere  to  be  suspended  so  that  the  axis  of  x  is  vertical,  and  let  / 
be  horizontal,  then  if  6  be  the  angle  which  the  axis  of  y  makes  with  the 
direction   of  /,  m  =  cos  6,  n=  —  sin  0,  and  the  expression  for  the   moment  becomes 

f  TT^T^hT}?    i'\  ^'«'  sin  2d, 

tending  to  increase  0.  The  axis  of  least  resistance  therefore  sets  axially,  but 
with  either  end  indifferently  towards  the  north. 

Since  in  all  bodies,  except  iron,  the  values  of  k  are  nearly  the  same  as  in 
a  vacuum,  the  coefficient  of  this  quantity  can  be  but  little  altered  by  changing 
the  value  of  k'  to  k,  the  value  in  space.     The  expression  then  becomes 

i^^^/Vsin2(9, 

independent  of  the  external  medium'". 


VII.     Permanent  magnetism  in  a  spherical  shell. 

The  case  of  a  homogeneous  shell  of  a  diamagnetic  or  paramagnetic  substance 
presents  no  difficulty.  The  intensity  within  the  shell  is  less  than  what  it  would 
have  been  if  the  shell  were  away,  whether  the  substance  of  the  shell  be  dia- 
magnetic or  paramagnetic.  When  the  resistance  of  the  shell  is  infinite,  and  when 
it  vanishes,  the  intensity  within  the  sheU  is  zero. 

In  the  case  of  no  resistance  the  entire  effect  of  the  shell  on  any  point, 
internal   or   external,   may   be   represented   by  supposing   a   superficial   stratum   of 

♦  Taking  the  more  general  case  of  magnetic  induction  referred  to  in  Art.  (28),  we  find,  in  the 
expression  for  the  moment  of  the  magnetic  forces,  a  constant  term  depending  on  T,  besides  those 
terms  which  dejjend  on  sines  and  cosines  of  6.  The  result  is,  that  in  every  complete  revolution  in 
the  negative  direction  round  the  axis  of  T,  a  certain  jMJsitive  amount  of  work  is  gained ;  but,  since 
no  inexhaustible  source  of  work  can  exist  in  nature,  we  must  admit  that  T-0  in  all  substances, 
with  resf>ect  to  magnetic  induction.  This  argument  does  not  hold  in  the  case  of  electric  conduction, 
or  in  the  case  of  a  body  through  which  heat  or  electricity  is  passing,  for  such  states  are  main- 
tained by  the  continual  expenditure  of  work.     See  Prof  Thomson,  Phil.  Mag.  March,  1851,  p.   186. 

28—2 


220  ON  Faraday's   lines  of  force. 

magnetic  matter  spread  over  the  outer  surface,  the  density  being  given  by  the 
equation 

p  =  3/  cos  d. 
Suppose  the   shell  now   to   be   converted   into   a   permanent  magnet,  so  that  the 
distribution  of  imaginary  magnetic   matter  is  invariable,  then  the  external  poten- 
tial due  to  the  shell  will  be 

p  =  —I—CO3  0, 

and  the  internal  potential  Pi—  ~ ^*' ^^^ 0. 

Now  let  us  investigate  the  eflfect  of  filling  up  the  shell  with  some  substance 
of  which  the  resistance  is  k,  the  resistance  in  the  external  medium  being  k". 
The  thickness  of  the  magnetized  shell  may  be  neglected.  Let  the  magnetic 
moment  of  the  permanent  magnetism  be  la^,  and  that  of  the  imaginary  super- 
ficial distribution  due  to  the  medium  k  =  Aa\     Then  the  potentials  are 

external  p'  =  {I-\-A)~  cos  6,        internal  ^,  =  (/+  ^ )  r  cos  0. 

The  distribution  of  real  magnetism  is  the  same  before  and  after  the  introduc- 
tion of  the  medium  k,  so  that 

l/+|/=i(/+4)+|(/+^), 

The  external  efiect  of  the  magnetized  shell  is  increased  or  diminished  according 
as  A;  is  greater  or  less  than  k'.  It  is  therefore  increased  by  filling  up  the  shell 
with  diamagnetic  matter,  and  diminished  by  filling  it  with  paramagnetic  matter, 
such  as  iron. 


VIII.     Electro-magnetic   spherical  shell. 

Let  us  take  as  an  example  of  the  magnetic  effects  of  electric  currents, 
an  electro-magnet  in  the  form  of  a  thin  spherical  sheU.  Let  its  radius  be  a, 
and  its  thickness  t,  and  let  its  external  effect  be  that  of  a  magnet  whose 
moment  is  /a*.  Both  within  and  without  the  shell  the  magnetic  effect  may  be 
represented  by   a  potential,   but   within   the   substance   of  the  shell,  where  there 


ON    FARADAY  S    LINES    OF    FORCE.  221 

are  electric  currents,  the  magnetic  effects  cannot  be  represented  by  a  potential. 
Let  p',  pi  be  the  external  and  internal  potentials, 

p'  =  1  -^cosd,        p^  =  Ar  cos  0, 

and  since  there  is  no  permanent  magnetism,  -^  =  -^- ,  when  r  =  a, 

A=-2L 

If  we  draw  any  closed  curve  cutting  the  shell  at  the  equator,  and  at  some 
other  point  for  which  0  is  known,  then  the  total  magnetic  intensity  round  this 
curve  will  be  Sla  cos  0,  and  as  this  is  a  measure  of  the  total  electric  current  which 
flows  through  it,  the  quantity  of  the  current  at  any  point  may  be  found  by 
differentiation.  The  quantity  which  flows  through  the  element  tcW  is  —  3/a  sin  0d0, 
so  that  the  quantity  of  the  current  referred  to  unit  of  area  of  section  is 

-3l^sm0. 
t 

If  the  shell  be  composed  of  a  wire  coiled  round  the  sphere  so  that  the  number 
of  coils  to  the  inch  varies  as  the  sine  of  0,  then  the  external  effect  will  be 
nearly  the  same  as  if  the  shell  had  been  made  of  a  uniform  conducting  sub- 
stance, and  the  currents  had  been  distributed  according  to  the  law  we  have  just 
given. 

If  a  wire  conducting  a  current  of  strength  /,  be  wound  round  a  sphere 
of  radius    a  so   that   the   distance   between  successive   coUs   measured  along  the 

2a 
axis  of  cc  is  — ,   then   there  wiU  be  n  coils   altogether,  and   the  value  of  /,  for 

the  resulting  electro-magnet  will  be 

The  potentials,  external  and  internal,  will  be 

P=I,Q^  003  0,  p,=  ■ 

The  interior  of  the  shell  is  therefore  a  uniform  magnetic  field. 


P  =I,Q  ^  cos^,         p,=  -21,-  -cos^. 


ON    FARADAY  S    LINES    OF    FORCE. 


IX.    Effect  of  the  core  of  the  electro-magnet. 

Now  let  us  suppose  a  sphere  of  diamagnetic  or  paramagnetic  matter  intro- 
duced into  the  electro-magnetic  coil.  The  result  may  be  obtained  as  in  the 
last  case,  and  the  potentials  become 

.,     J  n     Zk'     a?        ^  .J.  n      Sk      r 

The  external  effect  is  greater  or  less  than  before,  according  as  yfc'  is  greater 
or  less  than  k,  that  is,  according  as  the  interior  of  the  sphere  is  magnetic  or 
diamagnetic  with  respect  to  the  external  medium,  and  the  internal  effect  is 
altered  in  the  opposite  direction,  being  greatest  for  a  diamagnetic  medium. 

This  investigation  explains  the  effect  of  introducing  an  iron  core  into  an 
electro-magnet.  If  the  value  of  k  for  the  core  were  to  vanish  altogether,  the 
effect  of  the  electro-magnet  would  be  three  times  that  which  it  has  without 
the  core.     As  k  has  always  a  finite  value,  the  effect  of  the  core  is  less  than  this. 

In  the  interior  of  the  electro-magnet  we  have  a  uniform  field  of  magnetic 
force,  the  intensity  of  which  may  be  increased  by  surrounding  the  coil  with  a 
shell  of  iron.  If  k'  =  0,  and  the  shell  infinitely  thick,  the  effect  on  internal  points 
would  be  tripled. 

The  effect  of  the  core  is  greater  in  the  case  of  a  cylindric  magnet,  and 
greatest  of  aU  when  the  core  is  a  ring  of  soft  iron. 


X.    Electro-tonic  functions  in  spherical  dectro-magnet. 

Let  us  now  find  the  electro-tonic  functions  due  to  this  electro-magnet. 
They  will  be  of  the  form 

ao  =  0,  ^^  —  oiZ,  y^=  —<»y, 

where   tu   is  some  function  of  r.     Where  there  are  no  electric  currents,  we  must 
have  ttj,  6j,  Cj  each  =  0,  and  this  implies 

d    /_     .      doi\     ^ 

the  solution  of  which  is 


ON   Faraday's   lines   of   force.  223 

Within   the   shell    co  cannot  become  infinite ;  therefore   oi  =  C^  is  the  solution, 
and  outside  a  must  vanish  at  an  infinite  distance,  so  that 

is  the  solution  outside.     The  magnetic  quantity  within  the  shell  is  found  by  last 
article  to  be 

therefore  within  the  sphere 

Ln       1 


*  2a  3^  +  ^" 

Outside  the  sphere  we  must  determine  w  so  as  to  coincide  at  the  surface 
with  the  internal  value.     The  external  value  is  therefore 

=  _:?>        1       a' 
^         2a  3k  +  k'  r' ' 

where   the   shell   containing  the   currents  is   made   up   of    n    coils   of   wire,    con- 
ducting a  current  of  total  quantity  /j. 

Let  another  wire  be  coiled  round  the  shell  according  to  the  same  law,  and 
let  the  total  number  of  coils  be  n  ;  then  the  total  electro-tonic  intensity  EI^ 
round  the  second  coil  is  found  by  integrating 

EI^  =  I    (oa  sin  6ds, 


-i: 


along  the  whole  length  of  the  wire.     The  equation  of  the  wire  is 

/,      <^ 

cos  0  =  -y-  . 

nv 

where  n'  is  a  large  number;   and  therefore 

ds  =  a  sin  6d<^, 

=  —  ariTT  sin-  Odd, 

T?T      ^'"'       2  /  27r         ,j       1 

.*.  EI^=  -—  (oan  =  — —  ann  1 


3  """  ""       3  '"""^  3k  +  k" 
E  may  be  called  the  electro-tonic  coeflBcient  for  the  particular  wire. 


224  ON  Faraday's  lines  of  force. 


XI.     Spherical  electro-magnetic  CoU-Machine. 

We  have  now  obtained  the  electro-tonic  function  which  defines  the  action 
of  the  one  coil  on  the  other.  The  action  of  each  coil  on  itself  is  found  by- 
putting  n*  or  n*  for  nn\  Let  the  first  coil  be  connected  with  an  apparatus 
producing  a  variable  electro-motive  force  F.  Let  us  find  the  efiects  on  both 
wires,  supposing  their  total  resistances  to  be  i2  and  R,  and  the  quantity  of 
the  currents  /  and  /'. 

Let    N    stand    for    -^  (sk+k") '   *^^^  *^®   electro-motive  force   of   the  first 

wire  on  the  second  is 

dl 


That  of  the  second  on  itself  is 


Nnn    ,  . 
at 


-^<- 


The  equation  of  the  current  in  the  second  wire  is  therefore 

-iyr„n'f-iyr«-f=ij'i' (i). 

The  equation  of  the  current  in  the  first  wire  is 

-Nn'^^^-Nnn'§  +  F=RI. (2). 

EHminating  the  differential  coefficients,  we  get 

n        n'      ~  n* 

^^   ^[r^r]  di  +  ^-E^^RW (^)' 

from  which  to  find  /  and  F.    For  this   purpose   we   require  to  know  the  value 
of  i^  in  terms  of  t. 

Let  us  first  take  the  case  in  which  F  is  constant  and  /  and  T  initially  =  0. 
This  is  the  case  of  an  electro-magnetic  coil-machine  at  the  moment  when  the 
connexion  is  made  with  the  galvanic  trough. 


ON  Faraday's  lines  of  force.  225 

Putting  ^T  for  ^  [ji  + j^J  "^^  ^^ 

The  primary  current  increases  very  rapidly  from  0  to  >, ,  and  the  secondary 
commences  at  --jy  —  and  speedily  vanishes,  owing  to  the  value  of  t  being 
generally  very  small 

The  whole  work  done  by  either  current  in  heating  the  wire  or  in  any  other 
kind  of  action  is  found  from  the  expression 


PRdt. 


The  total  quantity  of  current  is 

^  Idt. 


f. 


For  the  secondary  current  we  find 


/; 


'-"-S;.       f."-m'r 


The   work   done  and  the   quantity   of  the  current  are  therefore  the   same  as 
if  a  current  of  quantity  F  =  —jrr-  had  passed  through  the  wire  for  a  time  t,  where 


--(^a- 


This  method  of  considering  a  variable  current  of  short  duration  is  due  to 
Weber,  whose  experimental  methods  render  the  determination  of  the  equivalent 
current  a  matter  of  great  precision. 

Now  let  the  electro-motive  force  F  suddenly  cease  while  the  current  in  the 
primary  wire  is  /<,  and  in  the  secondary  =  0.  Then  we  shall  have  for  the  subse- 
quent time 

,     .  -^  „     /„    Rn    -f 


226  ON  fahaday's  lines  of  force. 

R  n 
The  equivalent  currents  are  ^I^  and  ^I^  -^  — ,  and  their  duration  is  t. 

When  the  communication  with  the  source  of  the  current  is  cut  off,  there 
will  be  a  change  of  E.  This  will  produce  a  change  in  the  value  of  t,  so  that 
if  i2  be  suddenly  increased,  the  strength  of  the  secondary  current  will  be  increased, 
and  its  duration  diminished.  This  is  the  case  in  the  ordiaaiy  coU-machines.  The 
quantity  N  depends  on  the  form  of  the  machine,  and  may  be  determined  by 
experiment  for  a  machine  of  any  shape. 


XII.    Spherical  shell  revolving  in  magnetic  field. 

Let  us  next  take  the  case  of  a  revolving  shell  of  conducting  matter  under 
the  influence  of  a  uniform  field  of  magnetic  force.  The  phenomena  are  explained 
by  Faraday  in  his  Experimental  Researches,  Series  ii.,  and  references  are  there 
given  to  previous  experiments. 

Let  the  axis  of  z  be  the  axis  of  revolution,  and  let  the  angular  velocity 
be  6).  Let  the  magnetism  of  the  field  be  represented  in  quantity  by  /,  inclined 
at  an  angle  6  to  the  direction  of  z,  in  the  plane  of  zx. 

Let  R  be  the  radius  of  the  spherical  sheU,  and  T  the  thickness.  Let  the 
quantities  Oj,  ^o*  yoj.he  the  electro-tonic  functions  at  any  point  of  space;  a^,  \,  c„ 
«i»  Aj  7i  symbols  of  magnetic  quantity  and  intensity;  a^,  h^,  c„  a,,  13,,  y,  of 
electric  quantity  and  intensity.     Let  p,  be  the  electric  tension  at  any  point, 


^'+*a.l 


(1). 


ON  Faraday's  lines  op  roRCE.  227 

The  expressions  for  a,,  ^„  y,  due  to  the  magnetifim  of  the  field  are 

^,  =  5,  +  2  (2  Bin  ^  -  a;  cos  ^), 

A^,  B,,  Co  being  constants;   and  the   velocities   of  the  particles  of  the  revolving 
sphere  are 

dx  dy  dz     ^ 


We  have  therefore  for  the  electro-motive  forces 

An  dt  4iT  2 


a>=-7Z-^=--  7^008^0)0;, 


_  1    d^o  I     I  n 

$,= P  =  —  -:—  7T  cos  uayy, 

^*         47r  dt  An  2  ^' 


1    /  . 


'  4n  dt  An  2 

Returning  to  equations  (1),  we  get 

^db,     dct\     dfii     <^y» 


j^  (db^  _dc,\d§,  _dy,^^ 
\dz      dy)      dz      dy       ' 

\dx      dz  I      dx      dz      An  2 

^  /da,  _  dbA  ^  ^  _  ^^  ^  q 
dy      dx)       '  ' 


^dy      dx)      dy      dx 

From  which  with  equation  (2)  we  find 

11/.. 
ttj  =  -  7-  -7-  -7  sin  C/a>; 
k  An  A 

h,  =  0, 

I    1    I   .    a 

C,  =  T  T-  T  Sin  U(OX, 

k  An  A 


p,  =  -  — -  loi  {(x*  +  2/*)  cos  ^  -  a:s  sin  $]. 


228  ON  Faraday's  lines  of  force. 

These  expressions  would  determine  completely  the  motion  of  electricity  in 
a  revolving  sphere  if  we  neglect  the  action  of  these  currents  on  themselves. 
They  express  a  system  of  circular  currents  about  the  axis  of  y,  the  quantity 
of  current  at  any  point  being  proportional  to  the  distance  from  that  axis. 
The   external   magnetic    effect    will    be   that    of  a    small  magnet   whose   moment 

is  jx—i    w/sin  6,  with  its  direction  along  the  axis  of  y,  so  that  the  magnetism  of 

the  field  would   tend  to  turn   it   back  to  the   axis   of  x*. 

The  existence  of  these  currents  will  of  course  alter  the  distribution  of 
the  electro-tonic  functions,  and  so  they  will  react  on  themselves.  Let  the 
final  result  of  this  action  be  a  system  of  currents  about  an  axis  in  the  plane 
of  xy  inclined  to  the  axis  of  x  at  an  angle  ^  and  producing  an  external  effect 
equal  to   that   of  a  magnet   whose   moment  is  FR^. 

The  magnetic  inductive  components  within  the  shell  are 
/i  sin  ^  —  2/' cos  ^  in  x, 
—  21'  sm(f>  in.  y, 
/i  cos  6  in  2, 

Each  of  these  would  produce  its  own  system  of  currents  when  the  sphere 
is  in  motion,  and  these  would  give  rise  to  new  distributions  of  magnetism, 
which,  when  the  velocity  is  uniform,  must  be  the  same  as  the  original  distri- 
bution, 

(Ii  sin  6  —  21'  cos  <l>)  in  x  produces  2  t^— r  ot  {I^  sin  6  —  2  J'  cos  (f>)  in  y, 

T 

(  —  2T  sin  <^)  in  y  produces  2         ,  m  (21'  sin  ^)  in  x  ; 

IiQoad  in  z  produces  no  currents. 

We  must  therefore  have   the  following  equations,    since    the   state   of  the   shell 
is  the  same  at  every  instant, 

T 

Lam 6- 2r  cos <f)  =  /,  sin ^ -^  — — y  (o2T sin 6 

T 

-  2/  sin  <^  =  -— T  oj  (/,  sin  ^-  2r  cos  <^), 

*  The  expression  for  p^  indicates  a  variable  electric  tension  in  the  shell,  so  that  cuirents  might 
be  collected  by  wires  touching  it  at  the  equator  and  poles. 


ON    FARADAY  8    LINES    OF    FORCE.  229 

-hence  cot  <^  =  -        j  w,     /  =  ^      , 5-,^^     /i  sin  6. 


7-©" 


To  understand  the  meanmg  of  these  expressions  let  us  take  a  particular  case. 

Let  the  axis  of  the  revolving  shell  be  vertical,  and  let  the  revolution  be 
from  north  to  west.  Let  /  be  the  total  intensity  of  the  terrestrial  magnetism, 
and  let  the  dip  be  d,  then  Ico3$  is  the  horizontal  component  in  the  direction 
of  magnetic  north. 

The   result   of   the   rotation   is   to    produce   currents   in   the   shell    about   an 

T 

axis  inclined  at   a  small   angle  =  tan"* ——rco  to  the   south  of  magnetic  west,  and 

the   external    effect   of  these    currents    is   the    same    as   that  of  a  magnet  whose 
moment  is 

i    ,        ^"^  i?7cos  d. 

The  moment  of  the  couple  due  to  terrestrial  magnetism  tending  to  stop  the 
rotation  is 

2i7rk  To) 


2       24tTrkY  +  Tq}* 


i?Pc08'^, 


and  the  loss  of  work  due  to  this  in  unit  of  time  is 
24:Trk  T(o' 


2       247r^?+Pa>' 


i?P  cos'  d. 


This  loss  of  work  is  made  up  by  an  evolution  of  heat  in  the  substance  of 
the  shell,  as  is  proved  by  a  recent  experiment  of  M.  Foucault  (see  Coniptefi 
Rendus,  XLi.  p.  450). 


[From  the  Transacti&M  of  the  Royal  Scottish  Society  of  Arts,  VoL  iv.  Part  rv.] 


IX.     Description  of  a  New  Form   of  the  Platometer,  an  Instrument  for 
measuring  the  Areas  of  Plane  Figures  drawn  on  Paper*. 


1.  The  measurement  of  the  area  of  a  plane  figure  on  a  map  or  plan  is  an 
operation  so  frequently  occurring  in  practice,  that  any  method  by  which  it  may 
be  easily  and  quickly  performed  is  deserving  of  attention.  A  very  able  expo- 
sition of  the  principle  of  such  instruments  will  be  found  in  the  article  on 
Planimeters  in  the   Reports   of  the   Juries  of  the  Great  Exhibition,   1851. 

2.  In  considering  the  principle  of  instruments  of  this  kind,  it  will  be  most 
convenient  to  suppose  the  area  of  the  figure  measured  by  an  imaginary  straight 
line,  which,  by  moving  parallel  to  itself,  and  at  the  same 

time   altering  in   length  to   suit    the    form    of    the    area, 
accurately  sweeps  it  out. 

Let  AZ  be  a  fixed  vertical  line,  APQZ  the  boundary 
of  the  area,  and  let  a  variable  horizontal  line  move 
parallel  to  itself  firom  A  to  Z,  so  as  to  have  its  extremi- 
ties, P,  M,  in  the  curve  and  in  the  fixed  straight  line. 
Now,  suppose  the  horizontal  line  (which  we  shall  caU  the 
generating  line)  to  move  from  the  position  PM  to  QNy 
MN  being  some  small  quantity,  say  one  inch  for  distinct- 
ness. During  this  movement,  the  generating  line  will 
have  swept  out  the  narrow  strip  of  the  surface,  PMNQ, 
which  exceeds  the  portion  PMNp  by  the  smaU  triangle  PQp, 

But  since  MN,  the  breadth  of  the  strip,  is  one  inch,  the  strip  will  contain 
as  many  square  inches  as  PM  is  inches   long;    so   that,   when    the    generating 


♦  Bead  to  the  Society,  22nd  Jan.  1855. 


ON    A    NEW    FORM    OF    THE    PLATOMETER. 


231 


line  descends  one  inch,  it  sweeps  out  a  number  of  square  inches  equal  to  the 
number  of  linear  inches  in  its  length. 

Therefore,  if  we  have  a  machine  with  an  index  of  any  kind,  which,  while 
the  generating  line  moves  one  inch  downwards,  moves  forward  as  many  degrees 
as  the  generating  line  is  inches  long,  and  if  the  generating  line  be  alternately 
moved  an  inch  and  altered  in  length,  the  index  will  mark 
the  number  of  square  inches  swept  over  during  the  whole 
operation.  By  the  ordinary  method  of  limits,  it  may  be 
shown  that,  if  these  changes  be  made  continuous  instead 
of  sudden,  the  index  will  still  measure  the  area  of  the 
curve  traced  by  the  extremity  of  the  generating  line. 

3.  When  the  area  is  bounded  by  a  closed  curve,  as 
ABDC,  then  to  determine  the  area  we  must  carry  the  tra- 
cing point  from  some  point  A  of  the  curve,  completely  round 
the  circumference  to  A  again.  Then,  while  the  tracing  point 
moves  from  A  to  C,  the  index  will  go  forward  and  mea- 
sure the  number  of  square  inches  in  ACRP,  and,  while  it 
moves  from  C  to  D,  the  index  will  measure  backwards  the 
square    inches    in    CRPD,  so  that  it  will   now  indicate    the 

square  inches  in  ACD.  Similarly,  during  the  other  part  of  the  motion  from 
D  to  B,  and  from  B  to  D,  the  part  DBA  will  be  measured;  so  that  when 
the  tracing  point  returns  to  D,  the  instrument  will  have  measured  the  area 
ACDB.  It  is  evident  that  the  whole  area  will  appear  positive  or  negative 
according  as  the  tracing  point  is  carried  round  in  the  direction  ACDB  or  ABDC. 

4.  We  have  next  to  consider  the  various  methods  of  communicating  the 
required  motion  to  the  index.  The  first  is  by  means  of  two  discs,  the  first 
having  a  flat  horizontal  rough  surface,  turning  on  a  vertical 
axis,  OQ,  and  the  second  vertical,  with  its  circumference  rest- 
ing on  the  flat  surface  of  the  first  at  P,  so  as  to  be  driven 
round  by  the  motion  of  the  first  disc.  The  velocity  of  the 
second  disc  will  depend  on  OP,  the  distance  of  the  point  of 
contact  from  the  centre  of  the  first  disc;  so  that  if  OP  be 
made  always  equal  to  the  generating  line,  the  conditions  of  the  instrument  will 
be  fulfilled. 

This  is   accomplished   by  causing   the   index-disc   to   slip  along  the  radius  of 


232  ON  A  NEW  FORM  OF  THE  PLATOMETER. 

the  horizontal  disc ;  so  that  in  working  the  instrument,  the  motion  of  the  index- 
disc  is  compounded  of  a  rolling  motion  due  to  the  rotation  of  the  first  disc, 
and  a  slipping  motion  due  to  the  variation  of  the  generating  line. 

5.  In  the  instrument  presented  by  Mr  Sang  to  the  Society,  the  first  disc  is 
replaced  by  a  cone,  and  the  action  of  the  instrument  corresponds  to  a  mathe- 
matical valuation  of  the  area  by  the  use  of  oblique  co-ordinates.  As  he  has 
himself  explained  it  very  completely,  it  will  be  enough  here  to  say,  that  the 
index-wheel  has  still  a  motion  of  slipping  as  well  as  of  rolling. 

6.  Now,  suppose  a  wheel  rolling  on  a  surface,  and  pressing  on  it  with  a 
weight  of  a  pound;  then  suppose  the  coefficient  of  friction  to  be  |,  it  will 
require  a  force  of  2  oz.  at  least  to  produce  shpping  at  all,  so  that  even  if  the 
resistance  of  the  axis,  &c.,  amounted  to  1  oz.,  the  rolling  would  be  perfect.  But 
if  the  wheel  were  forcibly  pulled  sideways,  so  as  to  slide  along  in  the  direction 
of  the  axis,  then,  if  the  friction  of  the  axis,  &c.,  opposed  no  resistance  to  the 
turning  of  the  wheel,  the  rotation  would  still  be  that  due  to  the  forward  motion ; 
but  if  there  were  any  resistance,  however  small,  it  would  produce  its  effect  in 
diminishing  the  amount  of  rotation. 

The  case  is  that  of  a  mass  resting  on  a  rough  surface,  which  requires  a 
great  force  to  produce  the  shghtest  motion;  but  when  some  other  force  acts 
on  it  and  keeps  it  in  motion,  the  very  smallest  force  is  sufficient  to  alter  that 
motion  in  direction. 

7.  This  effect  of  the  combination  of  slipping  and  rolling  has  not  escaped 
the  observation  of  Mr  Sang,  who  has  both  measured  its  amount,  and  shown  how 
to  eliminate  its  effect.  In  the  improved  instrument  as  constructed  by  him,  I 
believe  that  the  greatest  error  introduced  in  this  way  does  not  equal  the  ordi- 
nary errors  of  measurement  by  the  old  process  of  triangulation.  This  accuracy, 
however,  is  a  proof  of  the  excellence  of  the  workmanship,  and  the  smoothness 
of  the  action  of  the  instrument;  for  if  any  considerable  resistance  had  to  be 
overcome,  it  would  display  itself  in  the  results. 

8.  Having  seen  and  admired  these  instruments  at  the  Great  Exhibition  in 
1851,  and  being  convinced  that  the  combination  of  shpping  and  roUing  was  a 
drawback  on  the  perfection  of  the  instrument,  I  began  to  search  for  some 
arrangement  by  which  the  motion  should  be  that  of   perfect   rolling    in    every 


ON  A  NEW  FORM  OF  TUE  PLATOMETER.  233 

motion  of  which  the  instrument  is  capable.  The  forms  of  the  rolUng  parts  which 
I  considered  were — 

1.  Two  equal  spheres. 

2.  Two  spheres,  the  diameters  being  as  1  to  2. 

3.  A  cone  and  cylinder,  axes  at  right  angles. 

Of  these,  the  first  combination  only  suited  my  purpose.  I  devised  several  modes 
of  mounting  the  spheres  so  as  to  make  the  principle  available.  That  which  I 
adopted  is  borrowed,  as  to  many  details,  from  the  instruments  already  con- 
structed, so  that  the  originality  of  the  device  may  be  reduced  to  this  principle — 
The  abolition  of  sUpping  by  the  use  of  two  equal  spheres. 

9.  The  instrument  (Fig.  1)  is  mounted  on  a  frame,  which  rolls  on  the  two 
connected  wheels,  MM,  and  is  thus  constrained  to  travel  up  and  down  the 
paper,  moving  parallel  to  itself 

CH  is  a  horizontal  axis,  passing  through  two  supports  attached  to  the 
frame,  and  carrying  the  wheel  K  and  the  hemisphere  LAP.  The  wheel  K  rolls 
on  the  plane  on  which  the  instrument  travels,  and  communicates  its  motion  to 
the  hemisphere,  which  therefore  revolves  about  the  axis  AH  with  a  velocity 
proportional  to  that  with  which  the  instrument  moves  backwards  or  forwards. 

FCO  is  a  framework  (better  seen  in  the  other  figures)  capable  of  revolving 
about  a  vertical  axis,  Cc,  being  joined  at  C  and  c  to  the  frame  of  the  instru- 
ment. The  parts  CF  and  CO  are  at  right  angles  to  each  other  and  horizontal. 
The  part  CO  carries  with  it  a  ring,  SOS,  which  turns  about  a  vertical  axis  Oo. 
This  ring  supports  the  index-.sphere  Bh  by  the  extremities  of  its  axis  Ss,  just 
as  the  meridian  circle  carries  a  terrestrial  globe.  By  this  arrangement,  it  will 
be  seen  that  the  axis  of  the  sphere  is  kept  always  horizontal,  while  its  centre 
moves  so  as  to  be  always  at  a  constant  distance  from  that  of  the  hemisphere. 
This  distance  must  be  adjusted  so  that  the  spheres  may  always  remain  in  con- 
tact, and  the  pressure  at  the  point  of  contact  may  be  regulated  by  means  of 
springs  or  compresses  at  0  and  o  acting  in  the  direction  OC,  oc.  In  this  way 
the  rotation  of  the  hemisphere  is  made  to  drive  the  index-sphere. 

10.  Now,  let  us  consider  the  working  of  the  instrument.  Suppose  the  arm 
CE  placed  so  as  to  coincide  with  CD,  then  0,  the  centre  of  the  index-sphere 
will  be  in  the  prolongation  of  the  axis  HA.  Suppose  also  that,  when  in  this 
position,  the  equator  hB  of  the  index-sphere  is  in  contact  with  the  pole  A  of 
the   hemisphere.     Now,  let   the   arch   be   turned   into  the  position   CE  as  in  the 


234  ON    A    NEW    FORM    OF    THE    PLATOMETER. 

figure,  then  the  rest  of  the  framework  will  be  turned  through  an  equal  angle, 
and  the  index-sphere  will  roll  on  the  hemisphere  till  it  come  into  the  position 
represented  in  the  figure.  Then,  if  there  be  no  slipping,  the  arc  AP  =  BP,  and 
the  angle  ACF  =  BOP. 

Next,  let  the  instrument  be  moved  backwards  or  forwards,  so  as  to  turn 
the  wheel  Kk  and  the  hemisphere  LI,  then  the  index-sphere  will  be  turned 
about  its  axis  Ss  by  the  action  of  the  hemisphere,  but  the  ratio  of  their  veloci- 
ties will  depend  on  their  relative  positions.  If  we  draw  PQ,  PR,  perpendiculars 
from  the  point  of  contact  on  the  two  axes,  then  the  angular  motion  of  the 
index-sphere  will  be  to  that  of  the  hemisphere,  as  PQ  is  to  PR;  that  is,  as 
PQ  is  to  QC,  by  the  equal  triangles  POQ,  PQC ;  that  is,  as  ED  is  to  DC, 
by  the  similar  triangles  CQP,  CDE. 

Therefore  the  ratio  of  the  angular  velocities  is  as  ED  to  DC,  but  since 
DC  is  constant,  this  ratio  varies  as  ED.  We  have  now  only  to  contrive  some 
way  of  making  ED  act  as  the  generating  line,  and  the  machine  is  complete 
(see  art.  2). 

11.  The  arm  CF  is  moved  in  the  following  manner: — Tt  is  a  rectangular 
metal  beam,  fixed  to  the  frame  of  the  instrument,  and  parallel  to  the  axis  AH. 
cEe  is  a  little  carriage  which  rolls  along  it,  having  two  rollers  on  one  side  and 
one  on  the  other,  which  is  pressed  against  the  beam  by  a  spring.  This  carriage 
carries  a  vertical  pin,  E,  turning  in  its  socket,  and  having  a  collar  above, 
through  which  the  arm  CF  works  smoothly.  The  tracing  point  G  is  attached 
to  the  carriage  by  a  jointed  frame  eGe,  which  is  so  arranged  that  the  point 
may  not  bear  too  heavily  on  the  paper. 

12.  When  the  machine  is  in  action,  the  tracing  point  is  placed  on  a  point 
in  the  boundary  of  the  figure,  and  made  to  move  round  it  always  in  one 
direction  till  it  arrives  at  the  same  point  again.  The  up-and-down  motion  of 
the  tracing  point  moves  the  whole  instrument  over  the  paper,  turns  the  wheel 
K,  the  hemisphere  LI,  and  the  index-sphere  Bh ;  while  the  lateral  motion  of 
the  tracing  point  moves  the  carriage  E  on  the  beam  Tt,  and  so  works  the  arm 
CF  and  the  framework  CO;  and  so  changes  the  relative  velocities  of  the  two 
spheres,  as  has  been  explained, 

13.  In  this  way  the  instrument  works  by  a  perfect  rolling  motion,  in  what- 
ever direction  the  tracing  point  is  moved;  but  since  the  accuracy  of  the  result 
depends   on  the  equality  of  the  arcs  AP  and  BP,  and  since  the  smallest  error 


ON    A    NEW    FORM    OF    THE     PLATOMETER.  235 

of  adjustment  would,  in  the  course  of  time,  produce  a  considerable  deviation 
from  this  equality,  some  contrivance  is  necessary  to  secure  it.  For  this  purpose 
a  wheel  is  fixed  on  the  same  axis  with  the  ring  SOs,  and  another  of  the  same 
size  is  fixed  to  the  frame  of  the  instrument,  with  its  centre  coinciding  with  the 
vertical  axis  through  C.  These  wheels  are  connected  by  two  pieces  of  watch- 
spring,  which  are  arranged  so  as  to  apply  closely  to  the  edges  of  the  wheels. 
The  first  is  firmly  attached  to  the  nearer  side  of  the  fixed  wheel,  and  to  the 
farther  side  of  the  moveable  wheel,  and  the  second  to  the  farther  side  of  the 
fixed  wheel,  and  the  nearer  side  of  the  moveable  wheel,  crossing  beneath  the 
first  steel  band.  In  this  way  the  spheres  are  maintained  in  their  proper  relative 
position;  but  since  no  instrument  can  be  perfect,  the  wheels,  by  preventing 
dei-angement,  must  cause  some  slight  slipping,  depending  on  the  errors  of  work- 
manship. This,  however,  does  not  ruin  the  pretensions  of  the  instrument,  for  it 
may  be  shown  that  the  error  introduced  by  slipping  depends  on  the  distance 
through  which  the  lateral  slipping  takes  place ;  and  since  in  this  case  it  must 
be  very  small  compared  with  its  necessarily  large  amount  in  the  other  instru- 
ments, the  error  introduced  by  it  must  be  diminished  in  the  same  proportion. 

14.  I  have  shewn  how  the  rotation  of  the  index-sphere  is  proportional  to 
the  area  of  the  figure  traced  by  the  tracing  point.  This  rotation  must  be 
measured  by  means  of  a  graduated  circle  attached  to  the  sphere,  and  read  oti" 
by  means  of  a  vernier.  The  result,  as  measured  in  degrees,  may  be  interpreted 
in  the  following  manner : — 

Suppose  the  instrument  to  be  placed  with  the  arm  CF  coinciding  with  CD, 
the  equator  Bh  of  the  index-sphere  touching  the  pole  A  of  the  hemisphere,  and 
the  index  of  the  vernier  at  zero :   then  let  these  four  operations  be   performed : — 

(1)  Let  the  tracing  point  be  moved  to  the  right  till  DE  =  DC,  and  there- 
fore DCE,  ACP,  and  F0B  =  A5\ 

(2)  Let  the  instrument  be  rolled  upwards  till  the  wheel  K  has  made  a 
complete  revolution,  carrying  the  hemisphere  with  it ;  then,  on  account  of  the 
equality  of  the  angles  SOP,  PC  A,  the  index-sphere  will  also  make  a  complete 
revolution. 

(3)  Let  the  arm  CF  be  brought  back  again  till  F  coincides  with  D. 

(4)  Let  the  instrument  be  rolled  back  again  through  a  complete  revolution 
of  the  wheel  K.  The  index-sphere  will  not  rotate,  because  the  point  of  contact 
is  at  the  pole  of  the  hemisphere. 


236  ON    A    NEW    FORM    OF    THE    PLATOMETER. 

The  tracing  point  has  now  traversed  the  boundary  of  a  rectangle,  whose 
length  is  the  circumference  of  the  wheel  A",  and  its  breadth  is  equal  to  CD; 
and  during  this  operation,  the  index-sphere  has  made  a  complete  revolution, 
360"  on  the  sphere,  therefore,  correspond  to  an  area  equal  to  the  rectangle  con- 
tained by  the  circumference  of  the  wheel  and  the  distance  CD.  The  size  of 
the  wheel  K  being  known,  different  values  may  be  given  to  CD,  so  as  to  make 
the  instrument  measure  according  to  any  required  scale.  This  may  be  done, 
either  by  shifting  the  position  of  the  beam  Tt,  or  by  having  several  sockets 
in  the  carriage  E  for  the  pin  which  directs  the  arm  to  work  in. 

15.  If  I  have  been  too  prolix  in  describing  the  action  of  an  instrument 
which  has  never  been  constructed,  it  is  because  I  have  myself  derived  great 
satisfaction  from  following  out  the  mechanical  consequences  of  the  mathematical 
theorem  on  which  the  truth  of  this  method  depends.  Among  the  other  forms 
of  apparatus  by  which  the  action  of  the  two  spheres  may  be  rendered  available, 
is  one  which  might  be  found  practicable  in  cases  to  which  that  here  given 
would  not  apply.  In  this  instrument  (Fig.  4)  the  areas  are  swept  out  by  a 
radius- vector  of  variable  length,  turning  round  a  fixed  point  in  the  plane.  The 
area  is  thus  swept  out  with  a  velocity  varying  as  the  angular  velocity  of  the 
radius-vector  and  the  square  of  its  length  conjointly,  and  the  construction  of  the 
machine  is  adapted  to  the  case  as  follows : — 

The  hemisphere  is  fixed  on  the  top  of  a  vertical  pillar,  about  which  the  rest 
of  the  instrument  turns.  The  index-sphere  is  supported  as  before  by  a  ring  and 
framework.  This  framework  turns  about  the  vertical  pillar  along  with  the  tra- 
cing point,  but  has  also  a  motion  in  a  vertical  plane,  which  is  communicated  to 
it  by  a  curved  slide  connected  with  the  tracing  point,  and  which,  by  means  of  a 
prolonged  arm,  moves  the  framework  as  the  tracing  point  is  moved  to  and  from 
the  pillar. 

The  form  of  the  curved  slide  is  such,  that  the  tangent  of  the  angle  of 
inclination  of  the  line  joining  the  centres  of  the  spheres  with  the  vertical  is 
proportional  to  the  square  of  the  distance  of  the  tracing  point  from  the  vertical 
axis  of  the  instrument.  The  curve  which  fulfils  this  condition  is  an  hyperbola, 
one  of  whose  asymptotes  is  vertical,  and  passes  through  the  tracing  point,  and 
the  other  horizontal  through  the  centre  of  the  hemisphere. 

The  other  parts  of  this  instrument  are  identical  with  those  belonging  to 
that  alreadv  described. 


VOL.  /.  PLATE  n. 


Fig.iFlan 


C  %K 


FigJF'runr  EleuaCion 


VOL.  I.  PLATE  U. 


Ti^.4. 


ON     A    NEW     FORM     OF    THE    PLATOMETER.  237 

When  the  tracing  point  is  made  to  traverse  the  boundary  of  a  plane  figure, 
there  is  a  continued  rotation  of  the  radius-vector  combined  with  a  change  of 
length.  The  rotation  causes  the  index-sphere  to  roll  on  the  fixed  hemisphere, 
while  the  length  of  the  radius-vector  determines  the  rate  of  its  motion  about  its 
axis,  so  that  its  whole  motion  measures  the  area  swept  out  by  the  radius-vector 
during  the  motion  of  the  tracing  point. 

The  areas  measured  by  this  instrument  may  either  lie  on  one  side  of  the 
pillar,  or  they  may  extend  all  round  it.  In  either  case  the  action  of  the 
instrument  is  the  same  as  in  the  ordinary  case.  In  this  form  of  the  instrument 
we  have  the  advantages  of  a  fixed  stand,  and  a  simple  motion  of  the  tracing 
point;  but  there  seem  to  be  difficulties  in  the  way  of  supporting  the  spheres 
and  arranging  the  shde ;  and  even  then  the  instrument  would  require  a  tall 
pillar,  in  order  to  take  in  a  large  area. 

16.  It  will  be  observed  that  I  have  said  little  or  nothing  about  the  prac- 
tical details  of  these  instruments.  Many  useful  hints  will  be  found  in  the  large 
work  on  Platometers,  by  Professor  T.  Gonnellu,  who  has  given  us  an  account 
of  the  difficulties,  as  well  as  the  results,  of  the  construction  of  his  most 
elaborate  instrument.  He  has  also  given  some  very  interesting  investio-ations 
into  the  errors  produced  by  various  irregularities  of  construction,  although,  as 
far  as  I  am  aware,  he  has  not  even  suspected  the  error  which  the  sliding  of 
the  index-wheel  over  the  disc  must  necessarily  introduce.  With  respect  to  this, 
and  other  points  relating  to  the  working  of  the  instrument,  the  memoir  of 
Mr  Sang,  in  the  Transactions  of  this  Society,  is  the  most  complete  that  I 
have  met  with.  It  may,  however,  be  as  well  to  state,  that  at  the  time  when 
I  devised  the  improvements  here  suggested,  I  had  not  seen  that  paper,  though 
I  had  seen  the  instrument  standing  at  rest  in  the  Crystal  Palace. 

Edinburgh,  30th  January,  1855, 
Note. — Since  the  design  of  the  above  instrument  was  submitted  to  the  Society  of  Arts, 
I  have  met  with  a  description  of  an  instrument  combining  simplicity  of  construction  with 
the  power  of  adaptation  to  designs  of  any  size,  and  at  the  same  time  more  portable  than 
any  other  instrument  of  the  kind.  Althougli  it  does  not  act  by  perfect  rolling,  and  there- 
fore belongs  to  a  different  class  of  instruments  from  that  described  in  this  paper,  I  think 
that  its  simplicity,  and  the  beauty  of  the  principle  on  which  it  acts,  render  it  worth  the 
attention  of  engineers  and  mechanists,  whether  practical  or  theoretical.  A  full  account  of 
this  instrument  is  to  be  found  in  Moigno's  "  Cosmos,"  5th  year,  Vol.  viii.,  Part  viii.,  p.  213, 
published  20th  February  1856.  Description  et  Theorie  du  planiniHre  polaire,  invents  par 
J.  Amsler,  de  Schaffuuse  en  Suisse. 
Cambridge,  30th  April,  1856. 


[From  the  Cambridge  Philosophical  Society  Proceedings,  Vol.  i.  pp.  173 — 175.] 


X.      0?i  the  Elementary  TJieory  of  Optical  Instruments. 

The  object  of  this  communication  was  to  shew  how  the  magnitude  and 
position  of  the  image  of  any  object  seen  through  an  optical  instrument  could 
be  ascertained  without  knowing  the  construction  of  the  instrument,  by  means 
of  data  derived  from  two  experiments  on  the  instrument.  Optical  questions 
are  generally  treated  of  with  respect  to  the  pencils  of  rays  which  pass  through 
the  instrument.  A  pencil  is  a  collection  of  rays  which  have  passed  through  one 
point,  and  may  again  do  so,  by  some  optical  contrivance.  Now  if  we  suppose 
all  the  points  of  a  plane  luminous,  each  will  give  out  a  pencil  of  rays,  and 
that  collection  of  pencils  which  passes  through  the  instrument  may  be  treated 
as  a  beam  of  hght.  In  a  pencil  only  one  ray  passes  through  any  point  of 
space,  unless  that  point  be  the  focus.  In  a  beam  an  infinite  number  of  rays, 
corresponding  each  to  some  point  in  the  luminous  plane,  passes  through  any 
point;  and  we  may,  if  we  choose,  treat  this  collection  of  rays  as  a  pencil 
proceeding  from  that  point.  Hence  the  same  beam  of  light  may  be  decomposed 
into  pencils  in  an  infinite  variety  of  ways;  and  yet,  since  we  regard  it  as  the 
same  collection  of  rays,  we  may  study  its  properties  as  a  beam  independently 
of  the  particular  way  in  which  we  conceive  it  analysed  into  pencils. 

Now  in  any  instrument  the  incident  and  emergent  beams  are  composed 
of  the  same  light,  and  therefore  every  ray  in  the  incident  beam  has  a 
corresponding  ray  in  the  emergent  beam.  We  do  not  know  their  path  within 
the  instrument,  but  before  incidence  and  after  emergence  they  are  straight 
lines,  and  therefore  any  two  points  serve  to  determine  the  direction  of  each. 

Let  us  suppose  the  instrument  such  that  it  forms  an  accurate  image  of  a 
plane  object  in  a  given  position.     Then  every  ray  which  passes  through  a  given 


ON    THE    ELEMENTARY    THEORY    OF    OPTICAL    INSTRUMENTS.  239 

point  of  the  object  before  incidence  passes  through  the  corresponding  point  of 
the  image  after  emergence,  and  this  determines  one  point  of  the  emergent  ray. 
If  at  any  other  distance  from  the  instrument  a  plane  object  has  an  accurate 
image,  then  there  will  be  two  other  corresponding  points  given  in  the  incident 
and  emergent  rays.  Hence  if  we  know  the  points  in  which  an  incident  ray 
meets  the  planes  of  the  two  objects,  we  may  find  the  incident  ray  by  joining 
the  points  of  the  two  images  corresponding  to  them. 

It  was  then  shewn,  that  if  the  image  of  a  plane  object  be  distinct,  flat,  and 
similar  to  the  object  for  two  different  distances  of  the  object,  the  image  of  any 
other  plane  object  perpendicular  to  the  axis  will  be  distinct,  flat  and  similar 
to  the  object. 

When  the  object  is  at  an  infinite  distance,  the  plane  of  its  image  is  the 
principal  focal  plane,  and  the  point  where  it  cuts  the  axis  is  the  piincipal 
focus.  The  line  joining  any  point  in  the  object  to  the  corresponding  point  of 
the  image  cuts  the  axis  at  a  fixed  point  called  the  focal  centre.  The  distance 
of  the  principal  focus  from  the  focal  centre  is  called  the  principal  focal  length, 
or  simply  the  focal  length. 

There  are  two  principal  foci,  etc.,  formed  by  incident  parallel  rays  passing 
in  opposite  directions  through  the  instrument.  If  we  suppose  light  always  to 
pass  in  the  same  direction  through  the  instrument,  then  the  focus  of  incident 
rays  when  the  emergent  rays  are  parallel  is  the  Jirst  principal  focus,  and  the 
focus  of  emergent  rays  when  the  incident  rays  are  parallel  is  the  second 
principal  focus. 

Corresponding  to  these  we  have  first  and  second  focal  centres  and  focal 
lengths. 

Now  let  Q,  be  the  focus  of  incident  rays,  P^  the  foot  of  the  perpendicular 
from  ^1  on  the  axis,  Q,  the  focus  of  emergent  rays,  P,  the  foot  of  the  corre- 
sponding perpendicular,  F^F^  the  first  and  second  principal  foci,  A^A^  the  first  and 
second  focal  centres,  then 

F\F\  _PjQr_FJP, 

A^Frp.QrFA.' 

lines  being  positive  when  measured  in  the  direction  of  the  light.  Therefore 
the  position  and  magnitude  of  the  image  of  any  object  is  found  by  a  simple 
proportion. 


240  ON    THE    ELEMENTARY    THEORY    OF    OPTICAL    INSTRUMENTS. 

In  one  important  class  of  instruments  there  are  no  principal  foci  or  focal 
centres.  A  telescope  in  which  parallel  rays  emerge  parallel  is  an  instance.  In 
such  instruments,  if  m  be  the   angular  magnifying  power,   the   linear  dimensions 

of  the  image  are   —  of  the  object,  and  the  distance  of  the  image  of  the  object 

from   the   image   of  the   object-glass  is   —^   of    the   distance    of   the    object    from 

the  object-glass.  Rules  were  then  laid  down  for  the  composition  of  instruments, 
and  suggestions  for  the  adaptation  of  this  method  to  second  approximations,  and 
the  method  itself  was  considered  with  reference  to  the  labours  of  Cotes,  Smith, 
Euler,  Lagrange,  and  Gauss  on  the  same  subject. 


[From  the  Report  of  the  British  Association,  1856.] 


XI.     On  a  Method  of  Drawing  the  Theoi-etical  Forms  of  Faraday  s  Lines  of 
Force  without  Calculation. 


The  method  applies  more  particularly  to  those  cases  in  which  the  lines 
are  entirely  parallel  to  one  plane,  such  as  the  lines  of  electric  currents  in  a 
thin  plate,  or  those  round  a  system  of  parallel  electric  currents.  In  such  cases, 
if  we  know  the  forms  of  the  lines  of  force  in  any  two  cases,  we  may  combine 
them  by  simple  addition  of  the  functions  on  which  the  equations  of  the  lines 
depend.  Thus  the  system  of  lines  in  a  uniform  magnetic  field  is  a  series  of 
parallel  straight  lines  at  equal  intervals,  and  that  for  an  infinite  straight  electric 
current  perpendicular  to  the  paper  is  a  series  of  concentric  circles  whose  radii 
are  in  geometric  progression.  Having  drawn  these  two  sets  of  lines  on  two 
separate  sheets  of  paper,  and  laid  a  third  piece  above,  draw  a  third  set  of  lines 
through  the  intersections  of  the  first  and  second  sets.  This  will  be  the  system 
of  lines  in  a  uniform  field  disturbed  by  an  electric  current.  The  most  interesting 
cases  are  those  of  uniform  fields  disturbed  by  a  small  magnet.  If  %ve  draw  a 
circle  of  any  diameter  with  the  magnet  for  centre,  and  join  those  points  in  which 
the  circle  cuts  the  lines  of  force,  the  straight  lines  so  drawn  will  be  parallel  and 
equidistant;  and  it  is  easily  shown  that  they  represent  the  actual  lines  of 
force  in  a  paramagnetic,  diamagnetic,  or  crystallized  body,  according  to  the 
nature  of  the  original  lines,  the  size  of  the  circle,  &c.  No  one  can  study 
Faraday's  researches  without  wishing  to  see  the  forms  of  the  Hnes  of  force. 
This  method,  therefore,  by  which  they  may  be  easily  drawn,  is  recommended 
to  the  notice  of  electrical  students. 


[From  the  Report  of  the  British  Association,  1856.] 


XII.     On  the   Unequal  Sensibility  of  the  Foramen  Centrale  to  Light  of 
different  Colours. 

When  observing  tlie  spectrum  formed  by  looking  at  a  long  vertical  slit 
through  a  simple  prism,  I  noticed  an  elongated  dark  spot  running  up  and  down 
in  the  blue,  and  following  the  motion  of  the  eye  as  it  moved  up  and  down 
the  spectrum,  but  refusing  to  pass  out  of  the  blue  into  the  other  colours.  It 
was  plain  that  the  spot  belonged  both  to  the  eye  and  to  the  blue  part  of  the 
spectrum.  The  result  to  which  I  have  come  is,  that  the  appearance  is  due  to 
the  yellow  spot  on  the  retina,  commonly  called  the  Foramen  Centrale  of  Soem- 
mering. The  most  convenient  method  of  observing  the  spot  is  by  presenting 
to  the  eye  in  not  too  rapid  succession,  blue  and  yellow  glasses,  or,  still  better, 
allowing  blue  and  yellow  papers  to  revolve  slowly  before  the  eye.  In  this  way 
the  spot  is  seen  in  the  blue.  It  fades  rapidly,  but  is  renewed  every  time  the 
yellow  comes  in  to  relieve  the  effect  of  the  blue.  By  using  a  Nicol's  prism 
along  with  this  apparatus,  the  brushes  of  Haidinger  are  well  seen  in  connexion 
with  the  spot,  and  the  fact  of  the  brushes  being  the  spot  analysed  by  polarized 
light  becomes  evident.  If  we  look  steadily  at  an  object  behind  a  series  of  bright 
bars  which  move  in  front  of  it,  we  shall  see  a  curious  bending  of  the  bars  as 
they  come  up  to  the  place  of  the  yellow  spot.  The  part  which  comes  over  the 
spot  seems  to  start  in  advance  of  the  rest  of  the  bar,  and  this  would  seem  to 
indicate  a  greater  rapidity  of  sensation  at  the  yellow  spot  than  in  the  surround- 
ing retina.  But  I  find  the  experiment  diflScult,  and  I  hope  for  better  results 
from  more  accurate  observers. 


[From  the  Report  of  the  British  Association,  1856.] 


XIII.     On  the  TJieory  of  Compound  Colours  with  reference  to  Mixtures  of  Blue 

and  Yellow  Light. 

When  we  mix  together  blue  and  yellow  paint,  we  obtain  green  paint.  This 
fact  is  well  known  to  all  who  have  handled  colours ;  and  it  is  universally 
admitted  that  blue  and  yellow  make  green.  Red,  yellow,  and  blue,  being  the 
primary  colours  among  painters,  green  is  regarded  as  a  secondary  colour,  arising 
from  the  mixture  of  blue  and  yellow.  Newton,  however,  found  that  the  green 
of  the  spectrum  was  not  the  same  thing  as  the  mixture  of  two  colours  of  the 
spectrum,  for  such  a  mixture  could  be  separated  by  the  prism,  while  the  green 
of  the  specti-um  resisted  further  decomposition.  But  still  it  was  believed  that 
yellow  and  blue  would  make  a  green,  though  not  that  of  the  spectrum.  As 
far  as  I  am  aware,  the  first  experiment  on  the  subject  is  that  of  M.  Plateau, 
who,  before  1819,  made  a  disc  with  alternate  sectors  of  prussian  blue  and  gam- 
boge, and  observed  that,  when  spinning,  the  resultant  tint  was  not  green,  but 
a  neutral  gray,  inclining  sometimes  to  yellow  or  blue,  but  never  to  green.  Prof 
J.  D,  Forbes  of  Edinburgh  made  similar  experiments  in  1849,  with  the  same 
result.  Prof  Helmholtz  of  Konigsberg,  to  whom  we  owe  the  most  complete 
investigation  on  visible  colour,  has  given  the  true  explanation  of  this  phaenomenon. 
The  result  of  mixing  two  coloured  powders  is  not  by  any  means  the  same  as 
mixing  the  beams  of  light  which  flow  from  each  separately.  In  the  latter  case 
we  receive  all  the  light  which  comes  either  from  the  one  powder  or  the  other. 
In  the  former,  much  of  the  light  coming  from  one  powder  falls  on  particles  of 
the  other,  and  we  receive  only  that  portion  which  has  escaped  absorption  by  one 
or  other.  Thus  the  light  coming  from  a  mixture  of  blue  and  yellow  powder, 
consists  partly  of  light  coming  directly  from  blue  particles  or  yellow  particles, 
and  partly  of  light  acted  on  by  both  blue  and  yellow  particles.  This  latter  light 
is  green,   since  the  blue  stops  the  red,  yellow,  and  orange,  and  the  yellow  stops 


244  ON     THE    THEORY     OF    COMPOUND    COLOURS. 

the  blue  and  violet  I  have  made  experiments  on  the  mixture  of  blue  and 
vellow  light — by  rapid  rotation,  by  con\bined  reflexion  and  transmission,  by  view- 
ing them  out  of  focus,  in  stripes,  at  a  gre;it  distiince,  by  throwing  the  colours 
of  the  spectrum  on  a  screen,  and  by  receiving  them  into  the  eye  directly  ;  and 
I  have  arranged  a  portable  apparatus  by  which  any  one  may  see  the  result  of 
this  or  any  other  mLxture  of  the  colours  of  the  spectrum.  In  all  these  cases 
blue  and  yellow  do  not  make  green.  I  have  also  made  experiments  on  the 
mixture  of  coloured  powders.  Those  which  I  used  principally  were  "mineral 
blue"  (from  copper)  and  "chrome-yellow."  Other  blue  and  yellow  pigments  gave 
curious  results,  but  it  was  more  difficult  to  make  the  mixtures,  and  the  greens 
were  less  uniform  in  tint.  The  mixtures  of  these  colours  were  made  by  weight, 
and  were  painted  on  discs  of  paper,  which  were  afterwards  treated  in  the  manner 
described  in  my  paper  "  On  Colour  as  perceived  by  the  Eye,"  in  the  Transactions 
of  the  Boyal  Soi.'icti/  of  Edinburgh,  Vol.  xxi.  Part  2.  The  \'isible  effect  of  the 
colour  is  estimated  in  terms  of  the  standard-coloured  papers : — vermilion  (V), 
ultramarine  (U),  and  emerald-green  (E).  The  accmucy  of  the  results,  and  their 
sijjnificance,  can  be  best  understood  by  referring  to  the  paper  before  mentioned. 
I  shall  denote  mineral  blue  by  B,  and  chrome-yellow  by  Y ;  and  B,  Y,  means 
a  mixture  of  three  parts  blue  and  five  parts  yellow. 

Given  Colour.  Standard  Colours.               Coefficient 

V.        U.  E.               of  brightness. 

B,           ,  100  =       2       36  7  45 

B-     Y,  ,  100  =       1       18  17  37 

B.     Y,  ,  100  =      4       11  34  49 

B,    Y,  ,  100  =9         5  40  54 

B,     Y.  ,  100  =     15         1  40  56 

B,     Y,  ,  100  =     22   -    2  44  64 

B,    Y.  ,  100  =    35-10  51  76 

B,     Y,  ,  100  =     64-19  64  109 

Y,  ,  100  =  180  -27  124  277 

The  columns  Y,  U,  E  give  the  proportions  of  the  standard  colours  which 
are  equivalent,  to  100  of  the  given  colour;  and  the  sum  of  V,  U,  E  gives  a  co- 
efficient, which  gives  a  general  idea  of  the  brightness.  It  will  be  seen  that  the 
tirst  admixture  of  yellow  diminishes  the  brightness  of  the  blue.  The  negative 
vidues  of  U  indicate  that  a  mixture  of  Y,  U,  and  E  cannot  be  made  equivalent 
to  the  given  colour.     The  experiments  from  which   these  results  were  taken   had 


ON  THE  THEORY  OF  COMPOUND  COLOURS.  245 

the  negative  values  tran-sferred  to  the  other  side  of  the  equation.  They  were 
all  made  by  means  of  the  colour-top,  and  were  verified  by  repetition  at  different 
times.  It  may  be  necessary  to  remark,  in  conclusion,  with  reference  to  the  mode 
of  registering  visible  colours  in  terms  of  three  arbitrary  standard  colours,  that  it 
proceeds  upon  that  theory  of  three  primary  elements  in  the  sensation  of  colour, 
which  treats  the  investigation  of  the  laws  of  visible  colour  as  a  bmnch  of  human 
physiology,  incapable  of  being  deduced  from  the  laws  of  light  itself,  as  set  forth 
in  physical  optics.  It  takes  advantage  of  the  methods  of  optics  to  study  vision 
itself;  and  its  appeal  is  not  to  physical  principles,  but  to  our  consciousness  of 
our  own  sensations. 


[From  the  Report  of  ike  British  Association,  1856.] 


XIV.     On  an  Instrument  to  illxLstrate  Poinsdt's  Theory  of  Rotation. 

In  studying  the  rotation  of  a  solid  body  according  to  Poinsdt's  method,  we 
have  to  consider  the  successive  positions  of  the  instantaneous  axis  of  rotation 
with  reference  both  to  directions  fixed  in  space  and  axes  assumed  in  the  moving 
body.  The  paths  traced  out  by  the  pole  of  this  axis  on  the  invariable  plane  and 
on  the  central  ellipsoid  form  interesting  subjects  of  mathematical  investigation. 
But  when  we  attempt  to  follow  with  our  eye  the  motion  of  a  rotating  body, 
we  find  it  difficult  to  determine  through  what  point  of  the  body  the  instantaneous 
axis  passes  at  any  time, — and  to  determine  its  path  must  be  still  more  difficult. 
I  have  endeavoured  to  render  visible  the  path  of  the  instantaneous  axis,  and  to 
vary  the  circumstances  of  motion,  by  means  of  a  top  of  the  same  kind  as  that 
used  by  Mr  Elliot,  to  illustrate  precession^'.  The  body  of  the  instrument  is  a 
hoUow  cone  of  wood,  rising  from  a  ring,  7  inches  in  diameter  and  1  inch  thick. 
An  iron  axis,  8  inches  long,  screws  into  the  vertex  of  the  cone.  The  lower 
extremity  has  a  point  of  hard  steel,  which  rests  in  an  agate  cup,  and  forms  the 
support  of  the  instrument.  An  iron  nut,  three  ounces  in  weight,  is  made  to 
screw  on  the  axis,  and  to  be  fixed  at  any  point;  and  in  the  wooden  ring  are 
screwed  four  bolts,  of  three  ounces,  working  horizontally,  and  four  bolts,  of  one 
ounce,  working  vertically.  On  the  upper  part  of  the  axis  is  placed  a  disc  of 
card,  on  which  are  drawn  four  concentric  rings.  Each  ring  is  divided  into  four 
quadrants,  which  are  coloured  red,  yellow,  green,  and  blue.  The  spaces  between 
the  rings  are  white.  When  the  top  is  in  motion,  it  is  easy  to  see  in  which  quad- 
rant the  instantaneous  axis  is  at  any  moment  and  the  distance  between  it  and 
the  axis  of  the  instrument;  and  we  observe, — 1st.  That  the  instantaneous  axis 
travels  in  a  closed  curve,  and  returns  to  its  original  position  in  the  body.     2ndly. 

*  Transactions  of  the  Royal  Scottish  Society  of  Arts,  1855. 


ON     AN     INSTRUMENT    TO    ILLUSTRATE    POINSOT  S    THEORY    OF    ROTATION.         247 

That  by  working  the  vertical  bolts,  we  can  make  the  axis  of  the  instrument 
the  centre  of  this  closed  curve.  It  will  then  be  one  of  the  principal  axes  of 
inertia.  3rdly.  That,  by  working  the  nut  on  the  axis,  we  can  make  the  order 
of  colours  either  red,  yellow,  green,  blue,  or  the  reverse.  When  the  order  of 
colours  is  in  the  same  direction  as  the  rotation,  it  indicates  that  the  axis  of  the 
instrument  is  that  of  greatest  moment  of  inertia.  4thly.  That  if  we  screw  the 
two  pairs  of  opposite  horizontal  bolts  to  different  distances  from  the  axis,  the 
path  of  the  instantaneous  pole  will  no  longer  be  equidistant  from  the  axis,  but 
will  describe  an  ellipse,  whose  longer  axis  is  in  the  direction  of  the  mean  axis 
of  the  instrument.  5thly.  That  if  we  now  make  one  of  the  two  horizontal  axes 
less  and  the  other  greater  than  the  vertical  axis,  the  instantaneous  pole  will 
separate  from  the  axis  of  the  instrument,  and  the  axis  will  incline  more  and  more 
till  the  spinning  can  no  longer  go  on,  on  account  of  the  obliquity.  It  is  easy 
to  see  that,  by  attending  to  the  laws  of  motion,  we  may  produce  any  of  the 
above  effects  at  pleasure,  and  illustrate  many  different  propositions  by  means  of 
the  same  instrument. 


[From  the  Transactions  of  the  Royal  Society  of  Edinburgh,  Vol.  xxi.  Part  iv.] 


XV.  On  a  Dynamical  Top,  for  exhibiting  the  phenomena  of  the  motion  of  a 
system  of  invariable  form  about  a  fixed  point,  with  some  suggestions  as  to 
the  Earth's  mx)tion. 

(Read  20th  April,  1857.) 

To  those  who  study  the  progress  of  exact  science,  the  common  spinning-top 
is  a  symbol  of  the  labours  and  the  perplexities  of  men  who  had  successfully 
threaded  the  mazes  of  the  planetary  motions.  The  mathematicians  of  the  last 
age,  searching  through  nature  for  problems  worthy  of  their  analysis,  foimd  in 
this  toy  of  their  youth,  ample  occupation  for  their  highest  mathematical  powers. 

No  illustration  of  astronomical  precession  can  be  devised  more  perfect  than 
that  presented  by  a  properly  balanced  top,  but  yet  the  motion  of  rotation  has 
intricacies  far  exceeding  those  of  the  theory  of  precession. 

Accordingly,  we  find  Euler  and  D'Alembert  devoting  their  talent  and  their 
patience  to  the  estabhshment  of  the  laws  of  the  rotation  of  solid  bodies. 
Lagrange  has  incorporated  his  own  analysis  of  the  problem  with  his  general 
treatment  of  mechanics,  and  since  his  time  M.  Poins6t  has  brought  the  subject 
under  the  power  of  a  more  searching  analysis  than  that  of  the  calculus,  in 
which  ideas  take  the  place  of  symbols,  and  intelligible  propositions  supersede 
equations. 

In  the  practical  department  of  the  subject,  we  must  notice  the  rotatory 
machine  of  Bohnenberger,  and  the  nautical  top  of  Troughton.  In  the  first  of 
these  instruments  we  have  the  model  of  the  Gyroscope,  by  which  Foucault  has 
been  able  to  render  visible  the  effects  of  the  earth's  rotation.  The  beautiful 
experiments  by  which  Mr  J.  EUiot  has  made  the  ideas  of  precession  so  familiar 
to  us  are  performed  with  a  top,  similar  in  some  respects  to  Troughton's,  though 
not  borrowed  from  his. 


ON    A     DYNAMICAL    TOP.  249 

The  top  which  I  have  the  honour  to  spin  before  the  Society,  differs  from 
that  of  Mr  Elliot  in  having  more  adjustments,  and  in  being  designed  to  exhibit 
far  more  complicated  phenomena. 

The  arrangement  of  these  adjustments,  so  as  to  produce  the  desired  effects, 
depends  on  the  mathematical  theory  of  rotation.  The  method  of  exhibiting  the 
motion  of  the  axis  of  rotation,  by  means  of  a  coloured  disc,  is  essential  to  the 
success  of  these  adjustments.  This  optical  contrivance  for  rendering  visible  the 
nature  of  the  rapid  motion  of  the  top,  and  the  practical  methods  of  applying 
the  theory  of  rotation  to  such  an  instrument  as  the  one  before  us,  are  the 
grounds  on  which  I  bring  my  instrument  and  experiments  before  the  Society 
as  my  own. 

I  propose,  therefore,  in  the  first  place,  to  give  a  brief  outline  of  such  parts 
of  the  theory  of  rotation  as  are  necessary  for  the  explanation  of  the  phenomena 
of  the  top. 

I  shall  then  describe  the  instrument  with  its  adjustments,  and  the  effect  of 
each,  the  mode  of  observing  of  the  coloured  disc  when  the  top  is  in  motion,  and 
the  use  of  the  top  in  illustrating  the  mathematical  theory,  with  the  method  of 
making  the  different  experiments. 

Lastly,  I  shall  attempt  to  explain  the  nature  of  a  possible  variation  in  the 
earth's  axis  due  to  its  figure.  This  variation,  if  it  exists,  must  cause  a  periodic 
inequality  in  the  latitude  of  every  place  on  the  earth's  surface,  going  through  its 
period  in  about  eleven  months.  The  amount  of  variation  must  be  very  small, 
but  its  character  gives  it  importance,  and  the  necessary  observations  are  already 
made,  and  only  require  reduction. 


On  the  Tlieory  of  Rotation. 

The  theory  of  the  rotation  of  a  rigid  system  is  strictly  deduced  from  the 
elementary  laws  of  motion,  but  the  complexity  of  the  motion  of  the  particles  of 
a  body  freely  rotating  renders  the  subject  so  intricate,  that  it  has  never  been 
thoroughly  understood  by  any  but  the  most  expert  mathematicians.  Many  who 
have  mastered  the  lunar  theory  have  come  to  erroneous  conclusions  on  this  sub- 
ject;  and  even  Newton  haa  chosen  to  deduce  the  disturbance  of  the  earth's  axis 
from  his  theory  of  the  motion  of  the  nodes  of  a  free  orbit,  rather  than  attack 
the  problem  of  the  rotation  of  a  solid  body. 


250  ON     A    DYNAMICAL    TOP. 

The  method  by  which  M.  Poinsot  has  rendered  the  theory  more  manageable, 
is  by  the  liberal  introduction  of  "appropriate  ideas,"  chiefly  of  a  geometrical 
character,  most  of  which  had  been  rendered  familiar  to  mathematicians  by  the 
writings  of  Monge,  but  which  then  first  became  illustrations  of  this  branch  of 
dynamics.  If  any  further  progress  is  to  be  made  in  simplifying  and  arranging 
the  theory,  it  must  be  by  the  method  which  Poins6t  has  repeatedly  pointed  out 
as  the  only  one  which  can  lead  to  a  true  knowledge  of  the  subject, — that  of 
proceeding  from  one  distinct  idea  to  another,  instead  of  trusting  to  symbols  and 
equations. 

An  important  contribution  to  our  stock  of  appropriate  ideas  and  methods  has 
lately  been  made  by  Mr  R.  B.  Hayward,  in  a  paper,  "On  a  Direct  Method  of 
estimatmg  Velocities,  Accelerations,  and  all  similar  quantities,  with  respect  to  axes, 
moveable  in  any  manner  in  Space."  {Trans.  Cambridge  Phil.  Soc.  Vol.  x.  Part  i.) 
*  In  this  communication  I  intend  to  confine  myself  to  that  part  of  the 
subject  which  the  top  is  intended  to  illustrate,  namely,  the  alteration  of  the 
position  of  the  axis  in  a  body  rotating  freely  about  its  centre  of  gravity.  I 
shall,  therefore,  deduce  the  theory  as  briefly  as  possible,  from  two  considera- 
tions only, — the  permanence  of  the  original  angular  momentum  in  direction  and 
magnitude,  and  the  permanence  of  the  original  vis  viva. 

•"'  The  mathematical  difiSculties  of  the  theory  of  rotation  arise  chiefly  from 
the  want  of  geometrical  illustrations  and  sensible  images,  by  which  we  might 
fix  the  results  of  analysis  in  our  minds. 

It  is  easy  to  understand  the  motion  of  a  body  revolving  about  a  fixed  axle. 
Every  point  in  the  body  describes  a  circle  about  the  axis,  and  returns  to  its 
original  position  after  each  complete  revolution.  But  if  the  axle  itself  be  in 
motion,  the  paths  of  the  different  points  of  the  body  will  no  longer  be  circular 
or  re-entrant.  Even  the  velocity  of  rotation  about  the  axis  requires  a  careful 
definition,  and  the  proposition  that,  in  all  motion  about  a  fixed  point,  there  is 
always  one  Hne  of  particles  forming  an  instantaneous  axis,  is  usually  given  in 
the  form  of  a  very  repulsive  mass  of  calculation.  Most  of  these  difficulties  may 
be  got  rid  of  by  devoting  a  little  attention  to  the  mechanics  and  geometry  of 
the  problem  before  entering  on  the  discussion  of  the  equations. 

Mr  Hayward,  in  his  paper  already  referred  to,  has  made  great  use  of  the 
mechanical  conception  of  Angular  Momentum. 

*  7th  May,  1857.     The  paragraphs  marked  thus  have  been  rewritten  since  the  paper  was  read. 


ON     A     DYNAMICAL    TOP.  251 

Definition. — Jlie  Angular  Momentum  of  a  particle  about  an  axis  is  mea- 
sured by  the  product  of  the  mass  of  the  particle,  its  velocity  resolved  in  the  normal 
plane,  and  the  perpendicular  from  the  axis  on  the  direction  of  motion. 

^'  The  angular  momentum  of  any  system  about  an  axis  is  the  algebraical 
sum  of  the  angular  momenta  of  its  parts. 

As  the  rate  of  change  of  the  linear  momentum  of  a  particle  measures  the 
moving  force  which  acts  on  it,  so  the  rate  of  change  of  angular  momentum 
measures  the  moment  of  that  force  about  an  axis. 

All  actions  between  the  parts  of  a  system,  being  pairs  of  equal  and  opposite 
forces,  produce  equal  and  opposite  changes  in  the  angular  momentum  of  those 
parts.  Hence  the  whole  angular  momentum  of  the  system  is  not  aflfected  by 
these  actions  and  re-actions. 

*  When  a  system  of  invariable  form  revolves  about  an  axis,  the  angular 
velocity  of  every  part  is  the  same,  and  the  angular  momentum  about  the  axis  is 
the  product  of  the  angular  velocity  and  the  moment  of  inertia  about  that  axis. 

*  It  is  only  in  particular  cases,  however,  that  the  whole  angular  momentum 
can  be  estimated  in  this  way.  In  general,  the  axis  of  angular  momentum  differs 
from  the  axis  of  rotation,  so  that  there  will  be  a  residual  angular  momentum 
about  an  axis  perpendicular  to  that  of  rotation,  imless  that  axis  has  one  of  three 
positions,  called  the  principal  axes  of  the  body. 

By  referring  everything  to  these  three  axes,  the  theory  is  greatly  simplified. 
The  moment  of  inertia  about  one  of  these  axes  is  greater  than  that  about  any 
other  axis  through  the  same  point,  and  that  about  one  of  the  others  is  a  mini- 
mum. These  two  are  at  right  angles,  and  the  third  axis  is  perpendicular  to 
their  plane,  and  is  called  the  mean  axis. 

*  Let  A,  B,  C  be  the  moments  of  inertia  about  the  principal  axes  through 
the  centre  of  gravity,  taken  in  order  of  magnitude,  and  let  Wj  oj.,  cd^  be  the 
angular  velocities  about  them,  then  the  angular  momenta  wHl  be  Ao)„  Bco. 
and  Cwj . 

Angular  momenta  may  be  compounded  like  forces  or  velocities,  by  the 
law  of  the  "parallelogram,"  and  since  these  three  are  at  right  angles  to  each 
other,  their  resultant  is 

JA^:^JTB%JTC^'  =  H (1), 

and   this  must   be   constant,    both   in   magnitude   and  direction  in  space,  since  no 
external  forces  act  on  the  body. 


252  ON    A    DYNAMICAL    TOP. 

We   shall   call   this   axis    of    angular  momentum   the   invariable  axis.      It  is 

perpendicular   to   what   has   been   called   the   invariable    plane.      Poins6t    calls    it 

the   axis   of  the  couple  of  impulsion.     The    direction-cosines   of    this   axis   in   the 

body  are, 

,     A(o,  B(o.  Ca)o 

«  =  ^,       m  =  -^,       ^  =  ^- 

Since  I,  m  and  n  vary  during  the  motion,  we  need  some  additional 
condition  to  determine  the  relation  between  them.  We  find  this  in  the  property 
of  the  vis  viva  of  a  system  of  invariable  form  in  which  there  is  no  friction. 
The  vis  viva  of  such  a  system  must  be  constant.  We  express  this  in  the 
equation 

Aoj,'  +  B(o,'+C(o,'=V (2). 

Substituting  the  values  of  Wi,  w^,  Wj  in  terms  of  I,  m,  n, 


Let   -i=a\       -T,  =  h\       ^=c\ 


=  e' 


A       '       B       '       C~   '       W 
and  this  equation  becomes 

a'Z'  +  6W  +  cV  =  e» (3), 

and    the    equation    to    the    cone,    described   by    the    invariable   axis   within   the 
body,  is 

(a'-e')x'  +  {h'-e')y'-\-{c'-e')z'  =  0 (4). 

The  intersections  of  this  cone  with  planes  perpendicular  to  the  principal 
axes  are  found  by  putting  x,  y,  or  z,  constant  in  this  equation.  By  giving 
e  various  values,  all  the  different  paths  of  the  pole  of  the  invariable  axis, 
corresponding  to  different  initial  circumstances,  may  be  traced. 

*In  the  figiu-es,  I  have  supposed  a' =  100,  6'=  107,  and  c"  =  110.  The 
first  figure  represents  a  section  of  the  various  cones  by  a  plane  perpendicular 
to  the  axis  of  x,  which  is  that  of  greatest  moment  of  inertia.  These  sections 
are  ellipses  having  their  major  axis  parallel  to  the  axis  of  h.  The  value  of  e* 
corresponding  to  each  of  these  curves  is  indicated  by  figures  beside  the  curve. 
The  ellipticity  increases  with  the  size  of  the  ellipse,  so  that  the  section 
corresponding  to  6^=107  would  be  two  parallel  straight  lines  (beyond  the  bounds 
of  the  figure),   after  which  the  sections  would  be  hyperbolas. 


ON     A     DYNAMICAL    TOP.  253 

*The  second  figure  represents  the  sections  made  by  a  plane,  perpendicular 
to  the  mean  axis.  They  are  all  hyperbolas,  except  when  6^=107,  when  the 
section  is  two  intersecting  straight  lines. 

The  third  figure  shows  the  sections  perpendicular  to  the  axis  of  least 
moment  of  inertia.  From  e'=110  to  ^"=107  the  sections  are  ellipses,  e*=107 
gives  two  parallel  straight  lines,  and  beyond  these  the  curves  are  hyperbolas. 

*The   fourth   and   fifth    figures    show    the    sections     of    the    series    of  cones 

made   by   a   cube   and   a   sphere   respectively.      The    use    of    these    figures  is    to 

exhibit  the  connexion  between  the  different  curves  described  about  the  three 
principal  axes  by  the  invariable  axis  during  the  motion  of  the  body. 

*We  have  next  to  compare  the  velocity  of  the  invariable  axis  with  respect 
to  the  body,  with  that  of  the  body  itself  round  one  of  the  principal  axes. 
Since  the  invariable  axis  is  fixed  in  space,  its  motion  relative  to  the  body 
must  be  equal  and  opposite  to  that  of  the  portion  of  the  body  through  which 
it  passes.  Now  the  angular  velocity  of  a  portion  of  the  body  whose  direction - 
cosines  are  I,  m,  n,  about  the  axis  of  x  is 

Substituting  the  values  of  w^,  w^,  w,,  in  terms  of  I,  m,  n,  and  taking 
account  of  equation  (3),  this  expression  becomes 

Changing  the  sign  and  putting  1=^tt  we  have  the  angular  velocity  of 
the  invariable  axis  about  that  of  x 

_    o>,     e'  —  a" 

always  positive  about  the  axis  of  greatest  moment,  negative  about  that  of  least 
moment,  and  positive  or  negative  about  the  mean  axis  according  to  the  value 
of  e*.  The  direction  of  the  motion  in  every  case  is  represented  by  the  arrows 
in  the  figures.  The  arrows  on  the  outside  of  each  figure  indicate  the  direction 
of  rotation  of  the  body, 

*If   we   attend   to   the   curve   described   by   the   pole   of  the   invariable   axis 


254  ON    A    DYNAMICAL    TOP. 

on  the  sphere  in  fig.    5,   we   shall   see    that   the    areas   described   by   that   point, 
if  projected  on  the  plane  of  yz,  are  swept  out  at  the  rate 

a" 
Now   the   semi-axes   of  the  projection   of  the   spherical   ellipse  described   by 
the  pole  are 


Dividing  the  area  of  this  ellipse  by  the  area  described  during  one  revo- 
lution of  the  body,  we  find  the  number  of  revolutions  of  the  body  during 
the  description  of  the  ellipse — 


The  projections  of  the  spherical  ellipses  upon  the  plane  of  yz  are  all 
similar  ellipses,  and  described  in  the  same  number  of  revolutions;  and  in  each 
ellipse  so  projected,  the  area  described  in  any  time  is  proportional  to  the 
number  of  revolutions  of  the  body  about  the  axis  of  x,  so  that  if  we  measure 
time  by  revolutions  of  the  body,  the  motion  of  the  projection  of  the  pole  of 
the  invariable  axis  is  identical  with  that  of  a  body  acted  on  by  an  attractive 
central  force  varying  directly  as  the  distance.  In  the  case  of  the  hyperbolas 
in  the  plane  of  the  greatest  and  least  axis,  this  force  must  be  supposed 
repulsive.  The  dots  in  the  figures  1,  2,  3,  are  intended  to  indicate  roughly 
the  progress  made  by  the  invariable  axis  during  each  revolution  of  the  body 
about  the  axis  of  x,  y  and  z  respectively.  It  must  be  remembered  that  the 
rotation  about  these  axes  varies  with  their  inclination  to  the  invariable  axis, 
so  that  the  angular  velocity  diminishes  as  the  inclination  increases,  and  there- 
fore the  areas  in  the  ellipses  above  mentioned  are  not  described  with  uniform 
velocity  in  absolute  time,  but  are  less  rapidly  swept  out  at  the  extremities  of 
the  major  axis  than  at  those  of  the  minor. 

*When  two  of  the  axes  have  equal  moments  of  inertia,  or  h  —  c,  then 
the  angular  velocity  (o^  is  constant,  and  the  path  of  the  invariable  axis  is 
circular,  the  number  of  revolutions  of  the  body  during  one  circuit  of  the 
invariable  axis,  being 


ON     A     DYNAMICAL    TOP.  255 

The  motion  is  in  the  same  direction  as  that  of  rotation,  or  in  the  opposite 
direction,  according  as  the  axis  of  x  is  that  of  greatest  or  of  least  moment 
of  inertia. 

*Both  in  this  case,  and  in  that  in  which  the  three  axes  are  unequal,  the 
motion  of  the  invariable  axis  in  the  body  may  be  rendered  very  slow  by 
dimlulshing  the  difference  of  the  moments  of  inertia.  The  angular  velocity  of 
the  axis  of  x  about  the  invariable  axis  in  space  is 

to. 


a'(l-l')' 

which  is  greater  or  less  than  Wj,  as  e*  is  greater  or  less  than  a\  and,  when 
these  quantities  are  nearly  equal,  is  very  nearly  the  same  as  Wj  itself.  This 
quantity  indicates  the  rate  of  revolution  of  the  axle  of  the  top  about  its 
mean  position,  and  is  very  easily  observed. 

*The  instantaneous  axis  is  not  so  easily  observed.  It  revolves  round  the 
invariable  axis  in  the  same  time  with  the  axis  of  x,  at  a  distance  which  Is  very 
small  in  the  case  when  a,  h,  c,  are  nearly  equal.  From  its  rapid  angular  motion 
in  space,  and  Its  near  coincidence  with  the  invariable  axis,  there  Is  no  advantage 
in  studying  its  motion  in  the  top. 

*By  making  the  moments  of  inertia  very  unequal,  and  in  definite  proportion 
to  each  other,  and  by  drawing  a  few  strong  lines  as  diameters  of  the  disc,  the 
combination  of  motions  will  produce  an  appearance  of  epicycloids,  which  are  the 
result  of  the  continued  intersection  of  the  successive  positions  of  these  lines,  and 
the  cusps  of  the  epicycloids  lie  in  the  curve  in  which  the  instantaneous  axis 
travels.     Some  of  the  figures  produced  in  this  way  are  very  pleasing. 

In  order  to  illustrate  the  theory  of  rotation  experimentally,  we  must  have 
a  body  balanced  on  its  centre  of  gravity,  and  capable  of  having  Its  principal 
axes  and  moments  of  inertia  altered  in  form  and  position  within  certain  limits. 
We  must  be  able  to  make  the  axle  of  the  instrument  the  greatest,  least,  or 
mean  principal  axis,  or  to  make  it  not  a  principal  axis  at  all,  and  we  must  be 
able  to  see  the  position  of  the  Invariable  axis  of  rotation  at  any  time.  There 
must  be  three  adjustments  to  regulate  the  position  of  the  centre  of  gravity, 
three  for  the  magnitudes  of  the  moments  of  inertia,  and  three  for  the  directions 
of  the  principal  axes,  nine  Independent  adjustments,  which  may  be  distributed 
as  we  please  among  the  screws  of  the  instrument. 


256  ON    A    DYNAMICAL    TOP. 

The  form  of  the  body  of  the  instrument  which  I  have  found  most  suitable  is 
that  of  a  bell  (p.  262,  fig.  6).  (7  is  a  hollow  cone  of  brass,  i2  is  a  heavy 
ring  cast  in  the  same  piece.  Six  screws,  with  heavy  heads,  x,  y,  z,  x,  y',  z, 
work  horizontally  in  the  ring,  and  three  similar  screws,  I,  m,  n,  work  vertically 
through  the  ring  at  equal  intervals.  AS  is  the  axle  of  the  instrument,  SS  is 
a  brass  screw  working  in  the  upper  part  of  the  cone  (7,  and  capable  of  being 
firmly  clamped  by  means  of  the  nut  c.  5  is  a  cylindrical  brass  bob,  which  may 
be  screwed  up  or  down  the  axis,  and  fixed  in  its  place  by  the  nut  7). 

The  lower  extremity  of  the  axle  is  a  fine  steel  point,  finished  without  emery, 
and  afterwards  hardened.  It  runs  in  a  little  agate  cup  set  in  the  top  of  the 
pillai'  P.  If  any  emery  had  been  embedded  in  the  steel,  the  cup  would  soon 
be  worn  out.  The  upper  end  of  the  axle  has  also  a  steel  point  by  which  it  may 
be  kept  steady  while  spinning. 

When  the  instrument  is  in  use,  a  coloured  disc  is  attached  to  the  upper 
end  of  the  axle. 

It  will  be  seen  that  there  are  eleven  adjustments,  nine  screws  in  the  brass 
ring,  the  axle  screwing  in  the  cone,  and  the  bob  screwing  on  the  axle.  The 
advantage  of  the  last  two  adjustments  is,  that  by  them  large  alterations  can  be 
made,  which  are  not  possible  by  means  of  the  small  screws. 

The  first  thing  to  be  done  with  the  instrument  is,  to  make  the  steel  point 
at  the  end  of  the  axle  coincide  with  the  centre  of  gravity  of  the  whole.  This 
is  done  roughly  by  screwing  the  axle  to  the  right  place  nearly,  and  then  balancing 
the  instrument  on  its  point,  and  screwing  the  bob  and  the  horizontal  screws  till 
the  instrument  will  remain  balanced  in  any  position  in  which  it  is  placed. 

When  this  adjustment  is  carefully  made,  the  rotation  of  the  top  has  no 
tendency  to  shake  the  steel  point  in  the  agate  cup,  however  irregular  the  motion 
may  appear  to  be. 

The  next  thing  to  be  done,  is  to  make  one  of  the  principal  axes  of  the 
central  ellipsoid  coincide  with  the  axle  of  the  top. 

To  effect  this,  we  must  begin  by  spinning  the  top  gently  about  its  axle, 
steadying  the  upper  part  with  the  finger  at  first.  If  the  axle  is  already  a 
principal  axis  the  top  will  continue  to  revolve  about  its  axle  when  the  finger  is 
removed.  If  it  is  not,  we  observe  that  the  top  begins  to  spin  about  some  other 
axis,  and  the  axle  moves  away  from  the  centre  of  motion  and  then  back  to  it 
again,  and  so  on,  alternately  widening  its  circles  and  contracting  them. 


ON    A    DYNAMICAL    TOP.  257 

It  is  impossible  to  observe  this  motion  successfully,  without  the  aid  of  the 
coloured  disc  placed  near  the  upper  end  of  the  axis.  This  disc  is  divided  into 
sectors,  and  strongly  coloured,  so  that  each  sector  may  be  recognised  by  its  colour 
when  in  rapid  motion.  If  the  axis  about  which  the  top  is  really  revolving,  falls 
within  this  disc,  its  position  may  be  ascertained  by  the  colour  of  the  spot  at  the 
centre  of  motion.  If  the  central  spot  appears  red,  we  know  that  the  invariable 
axis  at  that  instant  passes  through  the  red  part  of  the  disc. 

In  this  way  we  can  trace  the  motion  of  the  invariable  axis  in  the  revolving 
body,  and  we  find  that  the  path  which  it  describes  upon  the  disc  may  be  a  circle, 
an  ellipse,  an  hyperbola,  or  a  straight  line,  according  to  the  arrangement  of  the 
instrument. 

In  the  case  in  which  the  invariable  axis  coincides  at  first  with  the  axle  of 
the  top,  and  returns  to  it  after  separating  from  it  for  a  time,  its  true  path  is 
a  circle  or  an  ellipse  having  the  axle  in  its  circumference.  The  true  principal 
axis  is  at  the  centre  of  the  closed  curve.  It  must  be  made  to  coincide  with  the 
axle  by  adjusting  the  vertical  screws  I,  in,  n. 

Suppose  that  the  colour  of  the  centre  of  motion,  when  farthest  from  the 
axle,  indicated  that  the  axis  of  rotation  passed  through  the  sector  L,  then  the 
principal  axis  must  also  lie  in  that  sector  at  half  the  distance  from  the  axle. 

If  this  principal  axis  be  that  of  greatest  moment  of  inertia,  we  must  raise 
the  screw  I  in  order  to  bring  it  nearer  the  axle  A.  If  it  be  the  axis  of  least 
moment  we  must  lower  the  screw  /.  In  this  way  we  may  make  the  principal 
axis  coincide  with  the  axle.  Let  us  suppose  that  the  principal  axis  is  that  of 
greatest  moment  of  inertia,  and  that  we  have  made  it  coincide  with  the  axle  of 
the  instrument.  Let  us  also  suppose  that  the  moments  of  inertia  about  the 
other  axes  are  equal,  and  very  little  less  than  that  about  the  axle.  Let  the  top 
be  spun  about  the  axle  and  then  receive  a  disturbance  which  causes  it  to  spin 
about  some  other  axis.  The  instantaneous  axis  wiU  not  remain  at  rest  either 
in  space  or  in  the  body.  In  space  it  will  describe  a  right  cone,  completing  a 
revolution  in  somewhat  less  than  the  time  of  revolution  of  the  top.  In  the 
body  it  will  describe  another  cone  of  larger  angle  in  a  period  which  is  longer 
as  the  difierence  of  axes  of  the  body  is  smaller.'  The  invariable  axis  will  be 
fixed  in  space,  and  describe  a  cone  in  the  body. 

The  relation  of  the  different  motions  may  be  understood  from  the  following 
illustration.  Take  a  hoop  and  make  it  revolve  about  a  stick  which  remains  at 
rest  and  touches  the  inside  of  the  hoop.     The  section  of  the  stick  represents  the 


258  ON    A    DYNAinCAL    TOP. 

path  of  the  instantaneous  axis  in  space,  the  hoop  that  of  the  same  axis  in  the 
body,  and  the  axis  of  the  stick  the  invariable  axis.  The  point  of  contact  repre- 
sents the  pole  of  the  instantaneous  axis  itself,  travelling  many  times  round  the 
stick  before  it  gets  once  round  the  hoop.  It  is  easy  to  see  that  the  direction  in 
which  the  instantaneous  axis  travels  round  the  hoop,  is  in  this  case  the  same  as 
that  in  which  the  hoop  moves  round  the  stick,  so  that  if  the  top  be  spinning  in 
the  direction  i,  M,  N,  the  colours  will  appear  in  the  same  order. 

By  screwing  the  bob  B  up  the  axle,  the  difference  of  the  axes  of  inertia 
may  be  diminished,  and  the  time  of  a  complete  revolution  of  the  invariable 
axis  in  the  body  increased.  By  observing  the  number  of  revolutions  of  the  top 
in  a  complete  cycle  of  colours  of  the  invariable  axis,  we  may  determine  the 
ratio  of  the  moments  of  inertia. 

By  screwing  the  bob  up  farther,  we  may  make  the  axle  the  principal  axis  of 
least  moment  of  inertia. 

The  motion  of  the  instantaneous  axis  will  then  be  that  of  the  point  of 
contact  of  the  stick  with  the  outside  of  the  hoop  rolling  on  it.  The  order  of 
colours  will  be  N,  M,  L,  if  the  top  be  spinning  in  the  direction  Z,  M,  N,  and 
the  more  the  bob  is  screwed  up,  the  more  rapidly  will  the  colours  change,  till 
it  ceases  to  be  possible  to  make  the  observations  correctly. 

In  calculating  the  dimensions  of  the  parts  of  the  instrument,  it  is  necessary 
to  provide  for  the  exhibition  of  the  instrument  with  its  axle  either  the  greatest 
or  the  least  axis  of  inertia.  The  dimensions  and  weights  of  the  parts  of  the  top 
which  I  have  found  most  suitable,  are  given  in  a  note  at  the  end  of  this  paper. 

Now  let  us  make  the  axes  of  inertia  in  the  plane  of  the  ring  unequal.  We 
may  do  this  by  screwing  the  balance  screws  x  and  x^  farther  from  the  axle 
without  altering  the  centre  of  gravity. 

Let  us  suppose  the  bob  B  screwed  up  so  as  to  make  the  axle  the  axis  of 
least  inertia.  Then  the  mean  axis  is  parallel  to  xt^,  and  the  greatest  is  at  right 
angles  to  xdd^  in  the  horizontal  plane.  The  path  of  the  invariable  axis  on  the 
disc  is  no  longer  a  circle  but  an  ellipse,  concentric  with  the  disc,  and  having 
its  major  axis  parallel  to  the  mean  axis  xo^. 

The  smaller  the  difference  between  the  moment  of  inertia  about  the  axle  and 
about  the  mean  axis,  the  more  eccentric  the  ellipse  will  be;  and  if,  by  screwing 
the  bob  down,  the  axle  be  made  the  mean  axis,  the  path  of  the  invariable  axis 
will  be  no  longer  a  closed  curve,  but  an  hyperbola,  so  that  it  will  depart  alto- 
gether from  the  neighbourhood   of  the  axle.     When  the  top  is  in  this  condition 


ON    A    DYNAMICAL    TOP.  259 

it  must  be  spun  gently,  for  it  is  very  difficult  to  manage  it  when  its  motion 
gets  more  and  more  eccentric. 

When  the  bob  is  screwed  still  farther  down,  the  axle  becomes  the  axis  of 
greatest  inertia,  and  a:x^  the  least.  The  major  axis  of  the  ellipse  described  by 
the  invariable  axis  will  now  be  perpendicular  to  ccx",  and  the  farther  the  bob 
is  screwed  down,  the  eccentricity  of  the  ellipse  will  diminish,  and  the  velocity 
with  which  it  is  described  will  increase. 

I  have  now  described  all  the  phenomena  presented  by  a  body  revolving  freely 
on  its  centre  of  gravity.  If  we  wish  to  trace  the  motion  of  the  invariable  axis 
by  means  of  the  coloured  sectors,  we  must  make  its  motion  very  slow  compared 
■vvith  that  of  the  top.  It  is  necessary,  therefore,  to  make  the  moments  of  inertia 
about  the  principal  axes  very  nearly  equal,  and  in  this  case  a  very  small  change 
in  the  position  of  any  part  of  the  top  will  greatly  derange  the  'position  of  the 
principal  axis.  So  that  when  the  top  is  well  adjusted,  a  single  turn  of  one  of 
the  screws  of  the  ring  is  sufficient  to  make  the  axle  no  longer  a  principal  axis, 
and  to  set  the  true  axis  at  a  considerable  inclination  to  the  axle  of  the  top. 

All  the  adjustments  must  therefore  be  most  carefully  arranged,  or  we  may 
have  the  whole  apparatus  deranged  by  some  eccentricity  of  spinning.  The  method 
of  making  the  principal  axis  coincide  with  the  axle  must  be  studied  and  prac- 
tised, or  the  first  attempt  at  spinning  rapidly  may  end  in  the  destruction  of 
the  top,  if  not  of  the  table  on  which  it  is  spun. 


On  the  Earth's  Motion. 

We  must  remember  that  these  motions  of  a  body  about  its  centre  of  gra- 
vity, are  not  illustrations  of  the  theory  of  the  precession  of  the  Equinoxes. 
Precession  can  be  illustrated  by  the  apparatus,  but  we  must  arrange  it  so  that 
the  force  of  gravity  acts  the  part  of  the  attraction  of  the  sun  and  moon  in 
producing  a  force  tending  to  alter  the  axis  of  rotation.  This  is  easily  done  by 
bringing  the  centre  of  gravity  of  the  whole  a  little  below  the  point  on  which 
it  spins.  The  theory  of  such  motions  is  far  more  easily  comprehended  than 
that  which  we  have  been  investigating. 

But  the  earth  is  a  body  whose  principal  axes  are  unequal,  and  from  the 
phenomena  of  precession  we  can  determine  the  ratio  of  the  polar  and  equatorial 
axes  of  the  "central  ellipsoid;"  and  supposing  the  earth  to  have  been  set  in 
motion   about   any  axis   except   the   principal   axis,    or   to   have    had    its    original 


260  ON    A    DYNAMICAL    TOP. 

axis   disturbed   in   any   way,  its   subsequent   motion   would    be    that    of    the    top 
when  the  bob  is  a  little  below  the  critical  position. 

The  axis  of  angular  momentum  would  have  an  invariable  position  in  space, 
and  would  travel  with  respect  to  the  earth  round  the  axis  of  figure  with  a  velo- 

C—A 

city  =  0)  -— : —  where  w  is  the  sidereal  angular  velocity  of  the  earth.     The  apparent 

pole   of  the  earth   would   travel   (with   respect   to   the   earth)    from   west   to  east 

A 

round  the  true  pole,  completing  its  circuit  in  jy — ^  sidereal  days,  which  appears 

to  be  about  325*6  solar  days. 

The  instantaneous  axis  would  revolve  about  this  axis  in  space  in  about 
a  day,  and  would  always  be  in  a  plane  with  the  true  axis  of  the  earth  and 
the  axis  of  angular  momentum.  The  effect  of  such  a  motion  on  the  apparent 
position  of  a  star  would  be,  that  its  zenith  distance  would  be  increased  and 
diminished  during  a  period  of  325-6  days.  This  alteration  of  zenith  distance 
is  the  same  above  and  below  the  pole,  so  that  the  polar  distance  of  the  star 
is  unaltered.  In  fact  the  method  of  finding  the  pole  of  the  heavens  by  obser- 
vations of  stars,  gives  the  pole  of  the  invan-aUe  axis,  which  is  altered  only  by 
external  forces,  such  as  those  of  the  sun  and  moon. 

There  is  therefore  no  change  in  the  apparent  polar  distance  of  stars  due  to 
this  cause.  It  is  the  latitude  which  varies.  The  magnitude  of  this  variation 
cannot  be  determined  by  theory.  The  periodic  time  of  the  variation  may  be 
found  approximately  from  the  known  dynamical  properties  of  the  earth.  The 
epoch  of  maximum  latitude  cannot  be  found  except  by  observation,  but  it  must 
be  later  in  proportion  to  the  east  longitude  of  the  observatory. 

In  order  to  determine  the  existence  of  such  a  variation  of  latitude,  I  have 
examined  the  observations  of  Polaris  with  the  Greenwich  Transit  Circle  in  the 
years  1851-2-3-4.  The  observations  of  the  upper  transit  during  each  month  were 
collected,  and  the  mean  of  each  month  found.  The  same  was  done  for  the  lower 
transits.  The  difference  of  zenith  distance  of  upper  and  lower  transit  is  twice 
the  polar  distance  of  Polaris,  and  half  the  sum  gives  the  co-latitude  of  Greenwich. 

In  this  way  I  found  the  apparent  co-latitude  of  Greenwich  for  each  month 
of  the  four  years  specified. 

There  appeared  a  very  slight  indication  of  a  maximum  belonging  to  the  set 
of  months, 

March,  51.      Feb.  52.      Dec.  52.      Nov.  53.      Sept.  54. 


ON    A    DYNAMICAL    TOP.  261 

Tliis  result,  liowever,  is  to  be  regarded  as  very  doubtful,  as  there  did  not 
appear  to  be  evidence  for  any  variation  exceeding  half  a  second  of  space,  and 
more  observations  would  be  required  to  establish  the  existence  of  so  small  a 
variation  at  all. 

I  therefore  conclude  that  the  earth  has  been  for  a  long  time  revolving 
about  an  axis  very  near  to  the  axis  of  figure,  if  not  coinciding  with  it.  The 
cause  of  this  near  coincidence  is  either  the  original  softness  of  the  earth,  or 
the  present  fluidity  of  its  interior.  The  axes  of  the  earth  are  so  nearly  equal, 
that  a  considerable  elevation  of  a  tract  of  country  might  produce  a  deviation 
of  the  principal  axis  within  the  limits  of  observation,  and  the  only  cause  which 
would  restore  the  uniform  motion,  would  be  the  action  of  a  fluid  which  would 
gradually  diminish  the  oscillations  of  latitude.  The  permanence  of  latitude  essen- 
tially depends  on  the  inequality  of  the  earth's  axes,  for  if  they  had  been  all 
equal,  any  alteration  of  the  crust  of  the  earth  would  have  produced  new  prin- 
cipal axes,  and  the  axis  of  rotation  would  travel  about  those  axes,  altering  the 
latitudes  of  all  places,  and  yet  not  in  the  least  altering  the  position  of  the 
axis  of  rotation  among  the  stars. 

Perhaps  by  a  more  extensive  search  and  analysis  of  the  observations  of 
different  observatories,  the  nature  of  the  periodic  variation  of  latitude,  if  it  exist, 
may  be  determined.  I  am  not  aware  of  any  calculations  having  been  made  to 
prove  its  non-existence,  although,  on  dynamical  grounds,  we  have  every  reason 
to  look  for  some  very  small  variation  having  the  periodic  time  of  325-6  days 
nearly,  a  period  which  is  clearly  distinguished  from  any  other  astronomical  cycle, 
and  therefore  easily  recognised. 


262  ON    A    DYNAMICAL    TOP. 

NOTK 

Dimensions  and  Weights  of  the  parts  of  the  Dynamical  Top. 

I.     Body  of  the  top — 

Mean  diameter  of  ring,  4  inches. 
Section  of  ring,  |  inch  square. 

The  conical  portion  rises  from  the  upper  and  inner  edge  of  the  ring,  a 
height  of  1|  inches  from  the  base. 

The  whole  body  of  the  top  weighs 1  lb.  7    oz. 

Each   of  the   nine   adjusting   screws  has   its  screw   1    inch   long,  and   the 

screw  and  head  together  weigh  1  ounce.     The  whole  weigh         .        .  9    „ 

II.     Axle,  &c.— 

Length  of  axle  5  inches,  of  which  |  inch  at  the  bottom  is  occupied  by 
the  steel  point,  3J  inches  are  brass  with  a  good  screw  turned  on  it, 
and   the   remaining   inch   is  of  steel,  with  a  sharp  point  at  the  top. 

The  whole  weighs 1^  „ 

The  bob  B  has  a  diameter  of  1'4<  inches,  and  a  thickness  of  •4.     It  weighs  2|  „ 

The  nuts  b  and  c,  for  clamping  the  bob  and  the  body  of  the  top  on  the 

axle,  each  weigh  ^  oz. 1     „ 


Weight  of  whole  top  2  lb.  5J  oz. 

The  best  arrangement,  for  general  observations,  is  to  have  the  disc  of  card  divided 
into  four  quadrants,  coloured  with  vermilion,  chrome  yellow,  emerald  green,  and  ultramarine. 
These  are  bright  colours,  and,  if  the  vermilion  is  good,  they  combine  into  a  grayish  tint 
when  the  revolution  is  about  the  axle,  and  burst  into  brilliant  colours  when  the  axis  is 
disturbed.  It  is  useful  to  have  some  concentric  circles,  drawn  with  ink,  over  the  colours, 
and  about  12  radii  drawn  in  strong  pencil  lines.  It  is  easy  to  distinguish  the  ink  from 
the  pencil  lines,  as  they  cross  the  invariable  axis,  by  their  want  of  lustre.  In  this  way, 
the  path  of  the  invariable  axis  may  be  identified  with  great  accuracy,  and  compared  with 
theory. 


VOL.  I .   PLATE  III. 


riG  1 


FIG. 2 


PIG  4- 


VOL.  I .   PLATE  III 


riG    6 


[From  the  Philosophical  Magazine,  Vol.  xiv.] 

XVI.     Account  of  Experiments  on  the  Perception  of  Colour. 

To  the  Editors  of  the  Philosophical  Magazine  and  Journal. 

Gentlemen, 

The  experiments  which  I  intend  to  describe  were  undertaken  in  order 
to  render  more  perfect  the  quantitative  proof  of  the  theory  of  three  primary 
colours.  According  to  that  theory,  every  sensation  of  colour  in  a  perfect  human 
eye  is  distinguished  by  three,  and  only  three,  elementary  qualities,  so  that  in 
mathematical  language  the  quahty  of  a  colour  may  be  expressed  as  a  function 
of  three  independent  variables.  There  is  very  little  evidence  at  present  for 
deciding  the  precise  tints  of  the  true  primaries.  I  have  ascertained  that  a 
certain  red  is  the  sensation  wanting  in  colour-blind  eyes,  but  the  mathematical 
theory  relates  to  the  number,  not  to  the  nature  of  the  primaries.  If,  with  Sir 
David  Brewster,  we  assume  red,  blue,  and  yellow  to  be  the  primary  colours,  this 
amounts  to  saying  that  every  conceivable  tint  may  be  produced  by  adding 
together  so  much  red,  so  much  yellow,  and  so  much  blue.  This  is  perhaps  the 
best  method  of  forming  a  provisional  notion  of  the  theory.  It  is  evident  that  if 
any  colour  could  be  found  which  could  not  be  accurately  defined  as  so  much  of 
each  of  the  three  primaries,  the  theory  would  fall  to  the  ground.  Besides  this, 
the  truth  of  the  theory  requires  that  every  mathematical  consequence  of  assuming 
every  colour  to  be  the  result  of  mixture  of  three  primaries  should  also  be  true. 

I  have  made  experiments  on  upwards  of  100  diiferent  artificial  colours,  con- 
sisting of  the  pigments  used  in  the  arts,  and  their  mechanical  mixtures.  These 
experiments  were  made  primarily  to  trace  the  effects  of  mechanical  mixture  on 
various  coloured  powders ;  but  they  also  afford  evidence  of  the  truth  of  the 
theory,  that  all  these  various   colours   can  be   referred   to   three   primaries.      The 


264  EXPEKIMENTS  ON  THE  PERCEPTION  OF  COLOUR. 

following  experiments  relate  to  the  combinations  of  six  well-defined  colours  only, 
and  I  shall  describe  them  the  more  minutely,  as  I  hope  to  induce  those  who 
have  good  eyes  to  subject  them  to  the  same  trial  of  skill  in  distinguishing 
tints. 

The  method  of  performing  the  experiments  is  described  in  the  Transactions 
of  the  Royal  Society  of  Edinburgh,  Vol.  xxi.  Part  2.  The  colour- top  or  teetotum 
which  I  used  may  be  had  of  Mr  J.  M.  Bryson,  Edinburgh,  or  it  may  be  easily 
extemporized.  Any  rotatory  apparatus  which  will  keep  a  disc  revolving  steadily 
and  rapidly  in  a  good  light,  without  noise  or  disturbance,  and  can  be  easily 
stopped  and  shifted,  will  do  as  well  as  the  contrivance  of  the  spinning-top. 

The  essential  part  of  the  experiment  consists  in  placing  several  discs  of 
coloured  paper  of  the  same  size,  and  slit  along  a  radius,  over  one  another,  so 
that  a  portion  of  each  is  seen,  the  rest  being  covered  by  the  other  discs.  By 
sliding  the  discs  over  each  other  the  proportion  of  each  colour  may  be  varied, 
and  by  means  of  divisions  on  a  circle  on  which  the  discs  lie,  the  proportion  of 
each  colour  may  be  read  off.     My  circle  was  divided  into  100  parts. 

On  the  top  of  this  set  of  discs  is  placed  a  smaller  set  of  concentric  discs, 
so  that  when  the  whole  is  in  motion  round  the  centre,  the  colour  resulting  from 
the  mixture  of  colours  of  the  small  discs  is  seen  in  the  middle  of  that  arising 
from  the  laro-er  discs.  It  is  the  object  of  the  experimenter  to  shift  the  colours 
till  the  outer  and  inner  tints  appear  exactly  the  same,  and  then  to  read  off  the 
proportions. 

It  is  easy  to  deduce  from  the  theory  of  three  primary  colours  what  must 
be  the  number  of  discs  exposed  at  one  time,  and  how  much  of  each  colour  must 
appear. 

Every  colour  placed  on  either  circle  consists  of  a  certain  proportion  of  each 
of  the  primaries,  and  in  order  that  the  outer  and  inner  circles  may  have  precisely 
the  same  resultant  colour  in  every  respect,  there  must  be  the  same  amount  of 
each  of  the  primary  colours  in  the  outer  and  inner  circles.  Thus  we  have  as 
many  conditions  to  fulfil  as  there  are  primary  colours;  and  besides  these  we 
have  two  more,  because  the  whole  number  of  divisions  in  either  the  outer  or 
the  inner  circle  is  100,  so  that  if  there  are  three  primary  colours  there  wiU  be 
five  conditions  to  fulfil,  and  this  will  require  five  discs  to  be  disposable,  and 
these  must  be  arranged  so  that  three  are  matched  against  two,  or  four  against  one. 
If  we  take  six  difierent  colours,  we  may  leave  out  any  one  of  the  six,  and 
so   form  six   different   combinations   of    five    colours.      It   is   plain   that   these   six 


EXPERIMENTS  ON  THE  PERCEPTION  OF  COLOUR.  265 

combinations   must  be  equivalent   to   two   equations   only,    if  the    theory   of  three 
primaries  be  true. 

The  method  which  I  have  found  most  convenient  for  registering  the  result 
of  an  experiment,  after  an  identity  of  tint  has  been  obtained  in  the  inner  and 
outer  circles,  is  the  following : — 

Write  down  the  names  or  symbols  of  the  coloured  discs  each  at  the  top  of 
a  column,  and  underneath  write  the  number  of  degrees  of  that  colour  observed, 
calling  it  +  when  the  colour  is  in  the  outer  circle,  and  — when  it  is  in  the  inner 
circle ;  then  equate  the  whole  to  zero.  In  this  way  the  account  of  each  colour 
is  kept  in  a  separate  column,  and  the  equations  obtained  are  easily  combined  and 
reduced,  without  danger  of  confounding  the  colours  of  which  the  quantities  have 
been  measured.  The  following  experiments  were  made  between  the  3rd  and  11th 
of  September,  1856,  about  noon  of  each  day,  in  a  room  fronting  the  north, 
without  curtains  or  any  bright- coloured  object  near  the  window.  The  same 
combination  was  never  made  twice  in  one  day,  and  no  thought  was  bestowed 
upon  the  experiments  except  at  the  time  of  observation.  Of  course  the  gradua- 
tion was  never  consulted,  nor  former  experiments  referred  to,  till  each  combi- 
nation of  colours  had  been  fixed  by  the  eye  alone;  and  no  reduction  waa 
attempted  till  all  the  experiments  were  concluded. 

The  coloured  discs  were  cut  from  paper  painted  of  the  following  colours  :  — 
Vermilion,  Ultramarine,  Emerald-green,  Snow-white,  Ivory-black,  and  Pale 
Chrome-yellow.  They  are  denoted  by  the  letters  V,  U,  G,  W,  B,  Y  respectively. 
These  colours  were  chosen,  because  each  is  well  distinguished  from  the  rest,  so 
that  a  small  change  of  its  intensity  in  any  combination  can  be  observed.  Two 
discs  of  each  colour  were  prepared,  so  that  in  each  combination  the  colours  might 
occasionally  be  transposed  from  the  outer  circle  to  the  inner. 

The  first  equation  was  formed  by  leaving  out  vermilion.  The  remaining 
colours  are  Ultramarine-blue,  Emerald-green,  White,  Black,  and  Yellow.  We 
might  suppose,  that  by  mixing  the  blue  and  yellow  in  proper  proportions,  we 
should  get  a  green  of  the  same  hue  as  the  emerald-green,  but  not  so  intense, 
80  that  in  order  to  match  it  we  should  have  to  mix  the  green  with  white  to 
dilute  it,  and  with  black  to  make  it  darker.  But  it  is  not  in  this  way  that  we 
have  to  arrange  the  colours,  for  our  blue  and  yellow  produce  a  pinkish  tint,  and 
never  a  green,  so  that  we  must  add  green  to  the  combination  of  blue  and  yellow, 
to  produce  a  neutral  tint,  identical  with  a  mixture  of  white  and  black. 


266  EXPERIMENTS    ON    THE    PERCEPTION    OF    COLOUR. 

Blue,  green,  and  yellow  must  therefore  be  combined  on  the  large  discs,  and 
stand  on  one  side  of  the  equation,  and  black  and  white,  on  the  small  discs,  must 
stand  on  the  other  side.  In  order  to  facilitate  calculations,  the  colours  are 
always  put  down  in  the  same  order;  but  those  belonging  to  the  small  discs 
are  marked  negative.     Thus,  instead  of  writing 

54U  +  UG  +  32Y  =  32W  +  68B, 
we  write  +54U  + 14G-32W-68B  +  32Y  =  0. 

The  sum  of  all  the  positive  terms  of  such  an  equation  is  100,  being  the 
whole  number  of  divisions  in  tne  circle.  The  sum  of  the  negative  terms  is 
also  100. 

The  second  equation  consists  of  all  the  colours  except  blue ;  and  in  this 
way  we  obtain  six  different  combinations  of  five  colours. 

Each  of  these  combinations  was  formed  by  the  unassisted  judgment  of  my 
eye,  on  six  different  occasions,  so  that  there  are  thirty-six  independent  observa- 
tions of  equations  between  five  colours. 

Table  I.  gives  the  actual  observations,  with  their  dates. 

Table  II.  gives  the  result  of  summing  together  each  group  of  six  equations. 

Each  equation  in  Table  11.  has  the  sums  of  its  positive  and  negative  co- 
eflBcients  each  equal  to  600. 

Having  obtained  a  number  of  observations  of  each  combination  of  colours, 
we  have  next  to  test  the  consistency  of  these  results,  since  theoretically  two 
equations  are  sufficient  to  determine  all  the  relations  among  six  colours.  We 
must  therefore,  in  the  first  place,  determine  the  comparative  accuracy  of  the 
different  sets  of  observations.  Table  III.  gives  the  averages  of  the  errors  of 
each  of  the  six  groups  of  observations.  It  appears  that  the  combination  IV.  is 
the  least  accurately  observed,  and  that  VI.  is  the  best. 

Table  IV.  gives  the  averages  of  the  errors  in  the  observation  of  each  colour 
in  the  whole  series  of  experiments.  This  Table  was  computed  in  order  to  detect 
any  tendency  to  colour-blindness  in  my  own  eyes,  which  might  be  less  accurate 
in  discriminating  red  and  green,  than  in  detecting  variations  of  other  colours. 
It  appears,  however,  that  my  observations  of  red  and  green  were  more  accurate 
than    those   of    blue    or  yellow.      White   is   the   most   easily   observed,   from   the 


EXPERIMENTS     ON     THE    PERCEPTION     OF    COLOUR.  267 

brilliancy  of  the  colour,  and  black  is  liable  to  the  greatest  mistakes.  I  would 
recommend  this  method  of  examining  a  series  of  experiments  as  a  means  of 
detecting  partial  colour-blindness,  by  the  different  accuracy  in  observing  differ- 
ent colours.  The  next  operation  is  to  combine  all  the  equations  according  to 
their  values.  Each  was  first  multiplied  by  a  coefficient  proportional  to  its  ac- 
curacy, and  to  the  coefficient  of  white  in  that  equation.  The  result  of  adding 
all  the  equations  so  found  is  given  in  equation  (W). 

Equation  (Y)  is  the  result  of  similar  operations  with  reference  to  the 
yellow  on  each  equation. 

We  have  now  two  equations,  from  which  to  deduce  six  new  equations,  by 
eliminating  each  of  the  six  colours  in  succession.  We  must  first  combine  the 
equations,  so  as  to  get  rid  of  one  of  the  colours,  and  then  we  must  divide  by 
the  sum  of  the  positive  or  negative  coefficients,  so  as  to  reduce  the  equations 
to  the  form  of  the  observed  equations.  The  results  of  these  operations  are  given 
in  Table  V.,  along  with  the  means  of  each  group  of  six  observations.  It  will 
be  seen  that  the  differences  between  the  results  of  calculation  from  two  equations 
and  the  six  independent  observed  equations  are  very  small.  The  errors  in  red 
and  green  are  here  again  somewhat  less  than  in  blue  and  yellow,  so  that  there 
is  certainly  no  tendency  to  mistake  red  and  green  more  than  other  colours. 
The  average  difference  between  the  observed  mean  value  of  a  colour  and  the 
calculated  value  is  77  of  a  degree.  The  average  error  of  an  observation  in  any 
group  from  the  mean  of  that  group  was  '92.  No  observation  was  attempted 
to  be  registered  nearer  than  one  degree  of  the  top,  or  yo7  of  ^  circle  ;  so  that 
this  set  of  observations  agrees  with  the  theory  of  three  primary  colours  quite 
as  far  as  the  observations  can  warrant  us  in  our  calculations ;  and  I  think  that 
the  human  eye  has  seldom  been  subjected  to  so  severe  a  test  of  its  power  of 
distinguishing  colours.  My  eyes  are  by  no  means  so  accurate  in  this  respect  as 
many  eyes  I  have  examined,  but  a  little  practice  produces  great  improvement 
even  in  inaccurate  observers. 

I  have  laid  down,  according  to  Newton's  method,  the  relative  positions  of 
the  five  positive  colours  with  which  I  worked.  It  will  be  seen  that  W  lies 
within  the  triangle  VUG,  and  Y  outside  that  triangle. 

The  first  combination.  Equation  I.,  consisted  of  blue,  yellow,  and  green, 
taken  in  such  proportions  that  their  centre  of  gravity  falls  at  W, 


268 


EXPERIMENTS  ON  THE  PERCEPTION  OF  COLOUR. 


In  Equation  II.  a  mixture  of  red  and  green,  represented  in  the  diagram 
by  the  point  2,  is  seen  to  be  equivalent  to  a  mixture  of  white  and  yellow,  also 
represented  by  2,  which  is  a  pale  yellow  tint. 

Equation  III.  is  between  a  mixture  of  blue  and  yellow  and  another  of 
white  and  red.  The  resulting  tint  is  at  the  intersection  of  YU  and  WV ;  that 
is,  at  the  point  3,  which  represents  a  pale  pink  grey. 

Equation  IV.  is  between  VG  and  UY,  that  is,  at  4,  a  dirty  yellow. 

Equation  V.  is  between  a  mixture  of  white,  red,  and  green,  and  a  mixture 
of  blue  and  yellow  at  the  point  5,  a  pale  dirty  yellow. 

Equation  VI.  has  W.  for  its  resulting  tint. 


Blue,  U. 


Bed,  V 


G,  Green. 


Y,  Yellow. 

Of  all  the  resulting  tints,  that  of  Equation  IV.  is  the  furthest  from  white  ; 
and  we  find  that  the  observations  of  this  equation  are  affected  with  the  greatest 
errors.  Hence  the  importance  of  reducing  the  resultant  tint  to  as  nearly  a 
neutral  colour  as  possible. 

It  is  hardly  necessary  for  me  to  observe,  that  the  whole  of  the  numerical 
results  which  I  have  given  apply  only  to  the  coloured  papers  which  I  used, 
and  to  them  only  when  illuminated  by  daylight  from  the  north  at  mid-day  in 
September,  latitude  55".  In  the  evening,  or  in  winter,  or  by  candlelight,  the 
results  are  very  different.  I  believe,  however,  that  the  results  would  differ  far 
less  if  observed  by  different  persons,  than  if  observed  under  different  lights ; 
for  the  apparatus  of  vision  is  wonderfully  similar  in  different  eyes,  and  even  in 
colour-blind  eyes  the  system  of  perception  is  not  different,  but  defective. 


EXPERIMENTS    ON    THE    PERCEPTION    OF    COLOUR. 


269 


Table  I. — The  observations  arranged  in  groups. 


Equation  I. 

V  =  0. 

+  U. 

+  G. 

-W. 

-B.    +Y. 

Equation  IV. 

-V. 

+u. 

-O. 

w=o. 

+  B. 

+  Y. 

1856,  Sept.  3. 

0 

54 

12 

34 

66       34 

1856, 

Sept.  3. 

62 

15 

38 

0 

53 

32 

4. 

0 

58 

14 

31 

69       28 

4. 

63 

17 

37 

0 

46 

37 

5. 

0 

55 

12 

32 

68       33 

5. 

64 

16 

36 

0 

50 

34 

6. 

0 

54 

14 

32 

68       32 

6. 

62 

19 

38 

0 

46 

35 

8. 

0 

54 

14 

32 

68       32 

8. 

62 

19 

38 

0 

47 

34 

9. 

0 

53 

15 

32 

68       32 

9. 

63 

17 

37 

0 

49 

34 

Equation  n. 

-V. 

u=o. 

-G. 

+  \V. 

+  B.     +Y. 

Equation  V. 

+v. 

-U. 

+  G. 

+w. 

B  =  0. 

-Y. 

Sept.  3. 

59 

0 

41 

9 

71       20 

Sept.  3. 

56 

47 

28 

16 

0 

53 

4. 

61 

0 

39 

9 

68       23 

4. 

57 

50 

25 

18 

0 

50 

5. 

61 

0 

39 

9 

67       24 

5. 

66 

49 

24 

20 

0 

51 

6. 

59 

0 

41 

10 

66       24 

6. 

55 

47 

27 

18 

0 

53 

8. 

60 

0 

40 

9 

69       22 

8. 

54 

49 

26 

20 

0 

51 

9. 

61 

0 

39 

9 

68       23 

11. 

56 

50 

27 

17 

0 

50 

Equation  HI. 

+v. 

-u. 

G  =  0. 

+w. 

+  B.     -Y. 

Equation  VI. 

+v. 

+  U. 

+  G. 

-W. 

-B. 

Y  =  0. 

Sept.  3. 

20 

56 

0 

28 

52       44 

Sept.  3. 

38 

27 

35 

24 

76 

0 

4. 

23 

58 

0 

30 

47       42 

4. 

39 

27 

34 

24 

76 

0 

5. 

24 

56 

0 

29 

47       44 

5. 

40 

26 

34 

24 

76 

0 

6. 

20 

56 

0 

31 

49       44 

6. 

38 

28 

34 

24 

76 

0 

8. 

21 

57 

0 

29 

60       43 

8. 

39 

28 

33 

24 

76 

0 

9. 

21 

58 

0 

29 

50       42 

11. 

39 

27 

34 

23 

77 

0 

Table  II.— The  sums  of  the  observed  equations. 

V. 

U. 

G. 

W. 

B. 

Y. 

Equation    I. 

0 

+  328 

+    81 

-193 

- 

-407 

+ 

191 

II. 

_ 

361 

0 

-239 

+    55 

+  409 

+ 

136 

III. 

+ 

129 

-341 

0 

+  176 

+  295 

_ 

259 

IV. 

376 

+  103 

-224 

0 

+  291 

+ 

206 

V. 

+ 

334 

-292 

+  157 

+  109 

0 

- 

308 

VI. 

4 

233 

+  163 

+  204 

-143 

- 

-457 

0 

Table  III. — The  averages  of  the  errors  of  the  several  equations  from  the  means  expressed  in 

j^  parts  of  a  circle. 


Equations. 

I. 

n. 

m. 

IV. 

V. 

VL 

Errors. 

•94 

•85 

1-05 

117 

ro8 

•40 

Table  IV. — The  averages  of  the  errors  of  the  several  colours  from  the  means  in  y^  parts  of 

a  circle. 
Colours.  V.  D.  G.  W.  B,  Y. 

Errors.  -83  -99  •SO  -61  115  r09 

Average  error  on  the  whole  ^92. 

The  equations  from  which  the  reduced  results  were  obtained  were  calculated  as  follow : — 
Equation  for  (W)- (II)  +  2  (III)  +  (V)-2  (I) -4  (VI). 
Equation  for  (Y)  =  2  (I)  +  2  (II)  -  3  (III)  +  2  (IV)  -  3  (V> 


270  EXPERIMENTS    ON    THE    PERCEPTION    OF    COLOUR. 

These  operations  being  performed,  gave 

V.  U.  G.         W.         B.  Y. 

(W)      +    701  +  2282  +  1060-1474-3641  +  1072  =  0. 
(Y)      +2863-2761  +  1235  +  1131^    299-2767  =  0. 


From  these  were  obtained  the  following  results  by  elimination: — 

Table  V. 


Equation 

J   r  From  (W)  and  (Y) 
■  \  From  observation 

0 
0 

-54-1 
-54-7 

-13-9 
-13-5 

+  32-0 
+  32-1 

+  68-0 
+  67-9 

-32  0 
-31-8 

jj    (  From  (W)  and  (Y) 
*  ( From  observation 

-59-6 
-60-2 

0 
0 

-40-4 
-39-8 

+  10-4 
+    9-2 

+  66-0 
+  68-2 

+  23-6 
+  22-6 

,^^   f  From  (W)  and  (Y) 
\  From  observation 

-21-7 
-21-5 

+  57-4 
+  56-8 

0 
0 

-30-2 
-29-3 

-48-1 
-49-2 

+  42-6 
+  43-2 

f  From  (W)  and  (Y) 
(  From  observation 

-62-4 
-62-7 

+  18-6 
+  17-2 

-37-6 
-37-3 

0 
0 

+  45-7 
+  48-5 

+  35-7 
+  34-3 

1  From  (W)  and  (Y) 
■  (  From  observation 

+  55-6 
+  55-7 

-49-0 

-48-7 

+  25-2 
+  26-1 

+  19-2 
+  18-2 

0 
0 

-51-0 
-51-3 

^T  f  From  (W)  and  (Y)    -397      -26-6      -337      +227      +77-3  0 

^^•\  From  observation       -38-8     -27-2      -340      +28-3     +76-2  0 

James  Clerk  Maxwell. 
Glexlair,  Jum  13,  1857. 


[From    The   Quarterly   Journal  of  Pure  and   Applied   Mathematics,   Vol.    ii. 


XVII.     On  the  General  Laws  of  Optical  Instruments. 

The  optical  effects  of  compound  instruments  have  been  generally  deduced 
from  those  of  the  elementary  parts  of  which  they  are  composed.  The  formulae 
given  in  most  works  on  Optics  for  calculating  the  effect  of  each  spherical  sur- 
face are  simple  enough,  but,  when  we  attempt  to  carry  on  our  calculations  from 
one  of  these  surfaces  to  the  next,  we  arrive  at  fractional  expressions  so  com- 
phcated  as  to  make  the  subsequent  steps  very  troublesome. 

Euler  (Acad.  R.  de  Berlin,  1757,  1761.  Acad.  R.  de  Paris,  1765)  has  attacked 
these  expressions,  but  his  investigations  are  not  easy  reading.  Lagrange  (Acad. 
Berhn,  1778,  1803)  has  reduced  the  case  to  the  theory  of  continued  fractions 
and  so  obtained  general  laws. 

Gauss  [Dioptrische  Untersuchungen,  Gottingen,  1841)  has  treated  the  subject 
with  that  combination  of  analytical  skiU  with  practical  ability  which  he  displays 
elsewhere,  and  has  made  use  of  the  properties  of  principal  foci  and  principal 
planes.  An  account  of  these  researches  is  given  by  Prof.  Miller  in  the  third 
volume  of  Taylor's  Scientific  Memoirs.  It  is  also  given  entire  in  French  by 
M.  Bravais  in  Liouvilles  Journal  for  1856,  with  additions  by  the  translator. 

The  method  of  Gauss  has  been  followed  by  Prof  Listing  in  his  Treatise 
on  the  DioptHcs  of  the  Eye  (in  Wagner's  Handworterhuch  der  Physiologie)  from 
whom  I  copy  these  references,  and  by  Prof  Helmholtz  in  his  Treatise  vn 
Physiological   Optics   (in   Karsten's    Cyclopadie). 

The  earliest  general  investigations  are  those  of  Cotes,  given  in  Smith's 
Optics,  II.  76  (1738).  The  method  there  is  geometrical,  and  perfectly  general, 
but  proceeding  from  the  elementary  cases  to  the  more  complex  by  the  method 
of  mathematical  induction.  Some  of  his  modes  of  expression,  as  for  instance  his 
measure  of  "apparent  distance,"  have  never  come  into  use,  although  his  results 
may    easily    be    expressed    more    intelligibly ;    and    indeed    the    whole    fabric    of 


272  ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS. 

Geometrical  Optics,  as  conceived  by  Cotes  and  laboured  by  Smith,  has  fallen 
into  neglect,  except  among  the  writers  before  named.  Smith  tells  us  that  it 
was  with  reference  to  these  optical  theorems  that  Newton  said  "  If  Mr  Cotes 
had  lived  we  might  have  known  something." 

The  investigations  which  I  now  offer  are  intended  to  show  how  simple  and 
how  general  the  theory  of  instruments  may  be  rendered,  by  considering  the 
optical  effects  of  the  entire  instrument,  without  examining  the  mechanism  by 
which  those  effects  are  obtained.  I  have  thus  established  a  theory  of  "perfect 
instruments,"  geometrically  complete  in  itself,  although  I  have  also  shown,  that 
no  instrument  depending  on  refraction  and  reflexion,  (except  the  plane  mirror) 
can  be  optically  perfect.  The  first  part  of  this  theory  was  conununicated  to 
the  Philosophical  Society  of  Cambridge,  28th  April,  1856,  and  an  abstract  will 
be  found  in  the  Philosophical  Magazine,  November,  1856.  Propositions  VIII. 
and  IX.  are  now  added.     I  am  not  aware  that  the  last  has  been  proved  before. 

In  the  following  propositions  I  propose  to  establish  certain  rules  for  deter- 
mining, from  simple  data,  the  path  of  a  ray  of  light  after  passing  through  any 
optical  instrument,  the  position  of  the  conjugate  focus  of  a  luminous  point,  and 
the  magnitude  of  the  image  of  a  given  object.  The  method  which  I  shall  use 
does  not  require  a  knowledge  of  the  internal  construction  of  the  instrument  and 
derives  all  its  data  from  two  simple  experiments. 

There  are  certain  defects  incident  to  optical  instruments  from  which,  in  the 
elementary  theory,  we  suppose  them  to  be  free.  A  perfect  instrument  must 
fulfil  three  conditions : 

I.  Every  ray  of  the  pencil,  proceeding  from  a  single  point  of  the  object, 
must,  after  passing  through  the  instrument,  converge  to,  or  diverge  from,  a 
single  point  of  the  image.  The  corresponding  defect,  when  the  emergent  rays 
have  not  a  common  focus,  has  been  appropriately  called  (by  Dr  Whewell) 
Astigmatism. 

II.  If  the  object  is  a  plane  surface,  perpendicular  to  the  axis  of  the 
instrument,  the  image  of  any  point  of  it  must  also  lie  in  a  plane  perpendicular 
to  the  axis.  When  the  points  of  the  image  lie  in  a  curved  surface,  it  is  said 
to  have  the  defect  of  curvature. 

III.  The  image  of  an  object  on  this  plane  must  be  similar  to  the  object, 
whether  its  linear  dimensions  be  altered  or  not;  when  the  image  is  not  similar 
to  the  object,  it  is  said  to  be  distorted. 


ox    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS.  273 

An  image  free  from  these  three  defects  is  said  to  be  jycrfect. 

In  Fig.  1,  p.  285,  let  A^x^a^  represent  a  plane  object  perpendicular  to  the 
axis  of  an  instrument  represented  by  I.,  then  if  the  instrument  is  perfect,  as 
regards  an  object  at  that  distance,  an  image  A.a.p^_  will  be  formed  by  the 
emergent  rays,  which  will  have  the  following  properties  : 

I.  Every  ray,  which  passes  through  a  point  a^  of  the  object,  will  pass 
through   the    corresponding   point   a.   of  the    image. 

II.  Every  point  of  the  image  will  lie  in  a  plane  perpendicular  to  the  axis. 

III.  The   figure   A.ap^   will    be   similar  and   similarly  situated  to  the  figure 

Now  let  us  assume  that  the  instrument  is  also   perfect  as  regards  an  object 

in   the   plane   i?i?>,y8i   perpendicular   to   the   axis   through  -B„  and   that  the   image 

of  such   an   object   is   in   the   plane   B^fio    and    similar    to  the    object,    and   we 
shall  be  able  to  prove  the  following  proposition : 

Prop.  I.  If  an  instrument  give  a  perfect  image  of  a  plane  object  at  two 
different  distances  from  the  instrument,  all  incident  rays  having  a  common  focus 
will  have  a  common  focus  after  emergence. 

Let  Pj  be  the  focus  of  incident  rays.  Let  P-,a^^  be  any  incident  ray. 
Then,  since  every  ray  which  passes  through  a^  passes  through  a,,,  its  image  after 
emergence,  and  since  every  ray  which  passes  through  Z;,  passes  through  6,,  the 
direction  of  the  ray  P^a^\  after  emergence  must  be  ah.. 

Similarly,  since  a^  and  ySj  are  the  images  of  Oj  and  ^i,  if  P^a^^^  be  any 
other  ray,  its  direction  after  emergence  will  be  a„fi.y 

Join  a, a,,  h^^„  a.xL..,  hfi.,;  then,  since  the  parallel  planes  AjCt^a^  and  BJ}^, 
are  cut  by  the  plane  of  the  two  rays  through  P^,  the  intersections  cTiOi  and 
?jjSi  are  parallel. 

Also,  their  images,  being  similarly  situated,  are  parallel  to  them,  therefore 
a„a,  is  parallel  to  6^j,  and  the  lines  aJj„  and  a,^^  are  in  the  same  plane,  and 
therefore  either  meet  in  a  point  P^  or  are  parallel. 

Now  take  a  third  ray  through  P,,  not  in  the  plane  of  the  two  former. 
After  emergence  it  must  either  cut  both,  or  be  parallel  to  them.  If  it  cuts 
both  it  nuist  pass  through  the  point  P.,  and  then  every  other  ray  must  pass 
through  P.,  for  no  line  can  intersect  three  Hues,  not  in  one  plane,  without 
passing   through   their   point  of  intersection.     If  not,  then  all  the  emergent  rays 


274  ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS. 

are  parallel,  which  is  a  particular  case  of  a  perfect  pencil.  So  that  for  every 
position  of  the  focus  of  incident  rays,  the  emergent  pencil  is  free  from  astig- 
matism. 

Prop.  II.  In  an  instrument,  perfect  at  two  different  distances,  the  image 
of  any  plane  object  perpendicular  to  the  axis  will  be  free  from  the  defects  of 
curvature  and  distortion. 

Through  the  point  P,  of  the  object  draw  any  line  P,Q,  in  the  plane  of 
the  object,  and  through  P,Q,  draw  a  plane  cutting  the  planes  A„  B,  in  the  hnes 
ttio,,  h^,.  These  lines  will  be  parallel  to  P,Q,  and  to  each  other,  wherefore 
also  their  images,  a^o,,  b^„  will  be  parallel  to  P,Q,  and  to  each  other,  and 
therefore  in  one  plane. 

Now  suppose  another  plane  drawn  through  P^Q,  cutting  the  planes  A,  and 
B,  in  two  other  lines  parallel  to  P,Q^.  These  will  have  parallel  images  in  the 
planes  A^  and  B„  and  the  intersection  of  the  planes  passing  through  the  two 
pairs  of  images  wiU  define  the  line  P^Q,  which  will  be  parallel  to  them,  and 
therefore  to  P,Q„  and  will  be  the  image  of  P,Q,.  Therefore  P^,  the  image 
of  P,Qi  is  parallel  to  it,  and  therefore  in  a  plane  perpendicular  to  the  axis. 
Now  if  all  corresponding  lines  in  any  two  figures  be  parallel,  however  the  lines 
be  drawn,  the  figures  are  similar,  and  similarly  situated. 

From  these  two  propositions  it  follows  that  an  instrument  giving  a  perfect 
image  at  two  different  distances  will  give  a  perfect  image  at  all  distances.  We 
have  now  only  to  determine  the  simplest  method  of  finding  the  position  and 
magnitude  of  the  image,  remembering  that  wherever  two  rays  of  a  pencil  inter- 
sect, all  other  rays  of  the  pencil  must  meet,  and  that  aU  parts  of  a  plane 
object  have  their  images  in  the  same  plane,  and  equally  magnified  or  diminished. 

Prop.  III.  A  ray  is  incident  on  a  perfect  instrument  parallel  to  the  axis, 
to  find  its  direction  after  emergence. 

Let  a  J),  (fig.  2)  be  the  incident  ray,  A,a,  one  of  the  planes  at  which  an 
object  has  been  ascertained  to  have  a  perfect  image.  A,a,  that  image,  similar 
to  A^tti  but  in  magnitude  such  that  A/t^^xA.a,. 

Similarly  let  BJ),  be  the  image  of  BJj„  and  let  BM,  =  yBA-  Also  let 
A,B,  =  c,  and  A.X^  =  c^. 

Then  since  a,  and  h„  are  the  images  of  a,  and  \,  the  line  F^aK  will  be 
the   direction   of  the   ray  after   emergence,  cutting   the   axis   in   F^,  (unless  x  =  y. 


ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS.  275 

when    a.})^    becomes    parallel    to    the    axis).      The    point    F._    may   be    found,    by 
remembering  that  A^a,  =  B^b^,  Ajii  =  xAfL^,  B]j.  =  yDJj^.     We  find — 

■  "     'y-x 

Let  g^  be  the  point  at  which  the  emergent  ray  is  at  the  same  distance 
from  the  axis  as  the  incident  ray,  draw  gfi^  perpendicular  to  the  axis,  then 
we  have 

'   y-x 

Similarly,  if  aSiF^  be  a  ray,  which,  after  emergence,  becomes  parallel  to 
the  axis ;  and  gfi^  a  line  perpendicular  to  the  axis,  equal  to  the  distance  of 
the  parallel  emergent  ray,  then 

A,F,  =  c,-y~,        F,G,^^^^  . 
x—y  ^—y 

Definitions. 

I.  The  point  F^,  the  focus  of  incident  rays  when  the  emergent  rays  are 
parallel  to  the  axis,  is  called  the  Jirst  jprincii^al  focus   of  the   instrument. 

II.  The  plane  G^^  at  which  incident  rays  through  F^  are  at  the  same 
distance  from  the  axis  as  they  are  after  emergence,  is  called  the  first  princi- 
pal plane   of  the   instrument.     F^G^   is   called   the  first  focal   length. 

III.  The  point  F^,  the  focus  of  emergent  rays  when  the  incident  rays 
are  parallel,  is  called  the  second  principal  focus. 

IV.  The  plane  G,^.,  at  which  the  emergent  rays  are  at  the  same  distance 
from  the  axis,  as  before  incidence,  is  called  the  second  principal  plane,  and 
Ffi^  is  called  the  second  focal  length. 

When  x  =  y,  the  ray  is  parallel  to  the  axis,  both  at  incidence  and  emerg- 
ence, and  there  are  no  such  points  as  F  and  G.  The  instrument  is  then 
called  a   telescope.     x(  =  y)   is   called   the   linear  ina^nifying  power  and  is  denoted 

by  I,  and  the  ratio   -    is  denoted   by  n,    and  may  be  called  the  elongation. 

In  the  more  general  case,  in  which  x  and  y  are  different,  the  principal 
foci  and  principal  planes  afford  the  readiest  means  of  finding  the  position  of 
images. 


276  ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS. 

Prop.  IV.  Given  the  principal  foci  and  principal  planes  of  an  instrument, 
to   find   the   relations   of  the  foci   of  the   incident   and   emergent   pencils. 

Let  F„  F„  (fig.  3)  be  the  principal  foci,  G^,  G.,  the  principal  planes,  Q^ 
the  focus  of  incident  light,    Q^P^  perpendicular  to  the  axis. 

Through  ^1  draw  the  ray  Q^g^F^.  Since  this  ray  passes  through  F^  it 
emerges  parallel  to  the  axis,  and  at  a  distance  from  it  equal  to  G^g^.  Its 
direction  after  emergence  is  therefore  Q.,g^  where  G^g„  =  G^g^.  Through  Q^  draw 
Q{Yi  parallel  to  the  axis.  The  corresponding  emergent  ray  wiD  pass  through 
F^^,  and  will  cut  the  second  principal  plane  at  a  distance  G^y^_=  G-^y^,  so  that 
jP„y,  is  the  direction   of  this  ray  after  emergence. 

Since  both  rays  pass  through  the  focus  of  the  emergent  pencil,  Q^,  the 
point  of  intersection,  is  that  focus.  Draw  Q^P^  perpendicular  to  the  axis. 
Then   PxQi  =  G{Y^  =  G^y.,    and    G,g,  =  G^g^  =  P,Q.,.     By   similar   triangles  F,P,Q,  and 

■F.G^r 

P,F,  :  F,G,  ::  P,Q,  :  {G,g,  =  )  P,Q,. 
And  by  similar  triangles  F^P^Q^  and  F^G^y^ 

Pm  =  Gry^)  ■  P^Q^  ■■■■  ^^.  ■■  F^P^-- 
We  may  put  these  relations  into  the  concise  form 

P,F,_P,Q,_G^, 

F^r  p.Qr  F,p,' 

and  the  values  of  F„P^  and  PJ^^  are 

F  G    GJF  F  G 

F..P.=    '^'pf^"-  and  P.Q.  =  ^'P.Q,. 

These  expressions  give  the  distance  of  the  image  from  F^  measured  along  the 
axis,  and  also  the  perpendicular  distance  from  the  axis,  so  that  they  serve  to 
determine  completely  the  position  of  the  image  of  any  point,  when  the  princi- 
pal foci  and  principal  planes  are  known. 

Prop.  V.  To  find  the  focus  of  emergent  rays,  when  the  instrument  is  a 
telescope. 

Let  ^1  (fig.  4)  be  the  focus  of  incident  rays,  and  let  Q^aJ)^  be  a  ray 
parallel  to  the  axis ;  then,  since  the  instrument  is  telescopic,  the  emergent 
ray  Q^aM^  will  be  parallel  to  the  axis,  and  Q^P^^l.  Q^P^. 


ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS.  277 

Let    QiOiB^   be   a   ray    through   ^,,    the   emergent    ray    will    be    Q,a,J5,,    and 

AM,  ~  A,a,~  I.  A,a,  "  A.a,  "  A,B, ' 

so  that  -FT^  =  -4  r>'  =  n,  a  constant  ratio. 

P^B,     A,B^ 

Cor.     If  a   point    C  be   taken   on   the   axis   of  the    instrument   so    that 

^^^  =  A,B,-A^,  ^'^'  =  T:^ ^^^" 

then  CP,  =  n.CP,. 

Def.     The   point    C  is   called   the   centre   of  the   telescope. 

It  appears,  therefore,  that  the  image  of  an  object  in  a  telescope  has  its 
dimensions  perpendicular  to  the  axis  equal  to  I  times  the  corresponding  dimen- 
sions of  the  object,  and  the  distance  of  any  part  from  the  plane  through  C 
equal  to  n  times  the  distance  of  the  corresponding  part  of  the  object.  Of 
course  all  longitudinal  distances  among  objects  must  be  multipUed  by  n  to 
obtain   those   of  their   images,    and   the   tangent   of  the   angular  magnitude  of  an 

object    as    seen    from    a    given    point   in   the   axis   must   be   multipHed   by   -   to 

obtain   that   of  the   image   of  the   object   as   seen   from   the   image   of  the   given 

point.     The   quantity  -   is   therefore   called   the   angular    magnifying   power,   and 

is  denoted  by  m. 

Prop.  VI.  To  find  the  principal  foci  and  principal  planes  of  a  combina- 
tion of  two  instruments  having  a  common  axis. 

Let  /,  /'  (fig.  5)  be  the  two  instruments,  G^F^Ffi,  the  principal  foci  and 
planes  of  the  first,  G^F^F^G^  those  of  the  second,  V^<^^^S,  those  of  the  com- 
bination. Let  the  ray  g^jJj'g^  pass  through  both  instruments,  and  let  it  be 
parallel  to  the  axis  before  entering  the  fii'st  instrument.  It  will  therefore  pass 
through  F„  the  second  principal  focus  of  the  first  instrument,  and  through  g. 
so  that  G^^  =  (xi(7i. 

On  emergence  from  the  second  instrument  it  will  pass  through  ^^  the 
focus    conjugate    to    F,,    and    through    g^   in    the    second    principal   plane,  so  that 


278  ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS. 

(r.'g'  =  G^g^.  (f>i  is  by  definition  the  second  principal  focus  of  the  combination 
of  instruments,   and   if  T^y^   be   the   second   principal   plane,    then    r„y,  =  G^g^ 

We  have  now  to  find  the  positions  of  <f>,  and  Tj. 

By  Prop.  IV.,  we  have 

^^^== — F:Fr~  • 

Or,  tlie  distance  of  the  principal  focus  of  the  combination,  from  that  of  the 
second  instrument,  is  equal  to  the  product  of  the  focal  lengths  of  the  second 
instrument,  divided  by  the  distance  of  the  second  principal  focus  of  the  first 
instrument  from  the  first  of  the  second.     From  this  we  get 

r"jp'     jp'A     ^"'^^  {FjF^  —  F^G() 
Ctj  i^j  -  -t^2  9a  = jrpT , 

oi  G,<f>,  = jrp7 . 

Now,   by   the   pairs   of  similar   triangles   ^G^g^,  (jtV^y,  and  FJjr(g',   F^G^^, 

T,<j>,  _  r,y,  ^  %,  _  F„G, 

~g:4>.    Gig.    G:g(    g;f,- 

Multiplying  the  two  sides  of  the  former  equation  respectively  by  the  first  and 
last  of  these  equal  quantities,  we  get 

,      Gr^, .  GiF„' 

Or,  the  second  focal  distance  of  a  combination  is  the  product  of  the  second 
focal  lengths  of  its  two  components,  divided  by  the  distance  of  their  consecutive 
principal  foci. 

If  we  call  the  focal  distances  of  the  first  instrument  f^  and  /,,  those  of 
the  second  //  and  //,  and  those  of  the  combination  J\,  /j,  and  put  FJF^=d, 
then  the  positions  of  the  principal  foci  are  found  fi:om  the  values 

and  the  focal  lengths  of  the  combination  from 

'~    d     '  J'~    d    ' 


ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS.  279 

When  d  =  0,  all  these  values  become  infinite,  and  the  compound  instruiaent 
becomes  a  telescope. 

Prop.  VII.  To  find  the  linear  magnifying  power,  the  elongation,  and  the 
centre  of  the  instrument,  when  the  combination  becomes  a  telescope. 

Here  (fig.  6)  the  second  principal  focus  of  the  first  instrument  coincides  at  J' 
with  the  first  of  the  second.  (In  the  figure,  the  focal  distances  of  both  instru- 
ments are  taken  in  the  opposite  direction  from  that  formerly  assumed.  They  are 
therefore  to  be  regarded  as  negative.) 

In  the  first  place,  F,'  is  conjugate  to  F^,  for  a  pencil  whose  focus  before 
incidence  is  F^  will  be  parallel  to  the  axis  between  the  instruments,  and  will 
converge  to  i^/  after  emergence. 

Also  if  G^g^  be  an  object  in  the  first  principal  plane,  G,g„  will  be  its  first 
image,  equal  to  itself,  and  if  Hh  be  its  final  image 

^^^-       Gjr-~-  f:^ 

Now  the  linear  magnifying  power  is  7,-  ,  and  the  elongation  is  .'  . 
because  F.'  and  H  are  the  images   of  F.^   and    G^   respectively  ;    therefore 

l=-4^    and    n=££-. 
The  angular  magnifying   power  =  in  =  -=  —  4-7  • 
The   centre  of  the   telescope  is  at   the   point   C,   such   that 

When  n  becomes  1  the  telescope  has  no  centre.  The  efiect  of  the  Instruineni 
is  then  simply  to  alter  the  position  of  an  object  by  a  certain  distance  measured 
along  the  axis,  as  in  the  case  of  refraction  through  a  plate  of  glass  bounded  bv 
parallel  planes.  In  certain  cases  this  constant  distance  itself  disappears,  as  in 
the  case  of  a  combination   of  three   convex   lenses   of  which  the   focal    lengths  arr 


280  ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS. 

4,    1,   4  and  the  distances  4  and  4.     This  combination  simply  inverts  every  object 
without  altering  its  magnitude   or  distance  along   the  axis. 

The  preceding  theory  of  perfect  instruments  is  quite  independent  of  the 
mode  in  which  the  course  of  the  rays  is  changed  within  the  instrument,  as 
we  are  supposed  to  know  only  that  the  path  of  every  ray  is  straight  before 
it  enters,  and  after  it  emerges  from  the  instrument.  We  have  now  to  con- 
sider, how  far  these  results  can  be  applied  to  actual  instruments,  in  which 
the  course  of  the  rays  is  changed  by  reflexion  or  refraction.  "We  know  that 
such  instruments  may  be  made  so  as  to  fulfil  approximately  the  conditions  of 
a  perfect  instrument,  but  that  absolute  perfection  has  not  yet  been  obtained. 
Let  us  inquire  whether  any  additional  general  law  of  optical  instruments  can 
be  deduced  from  the  laws  of  reflexion  and  refraction,  and  whether  the  imper- 
fection of  instruments  is   necessary  or  removeable. 

The  following  theorem  is  a  necessary  consequence  of  the  known  laws  of 
reflexion  and  refraction,  whatever  theory  we  adopt. 

If  we  multiply  the  length  of  the  parts  of  a  ray  which  are  in  diflerent 
media  by  the  indices  of  refraction  of  those  media,  and  call  the  sum  of  these 
products  the  reduced  path  of  the  ray,  then  : 

I.  The  extremities  of  all  rays  from  a  given  origin,  which  have  the  same 
reduced  path,  lie  in  a  surface  normal  to  those  rays. 

II.  When  a  pencil  of  rays  is  brought  to  a  focus,  the  reduced  path  from 
the  origin  to  the  focus  is  the  same  for  every  ray  of  the  pencil. 

In  the  undulatory  theory,  the  "  reduced  path "  of  a  ray  is  the  distance 
through  which  light  would  travel  in  space,  during  the  time  which  the  ray 
takes  to  traverse  the  various  media,  and  the  surface  of  equal  "  reduced  paths " 
is  the  wave-surface.  In  extraordinary  refraction  the  wave-surface  is  not  always 
normal  to  the  ray,  but  the  other  parts  of  the  proposition  are  true  in  this  and  all 
other  cases. 

From  this  general  theorem  in  optics  we  may  deduce  the  following  propo- 
sitions, true  for  all  instruments  depending  on  refraction  and  reflexion. 

Prop.  VIII.  In  any  optical  instrument  depending  on  refraction  or  reflex- 
ion, if  ajtti,  />i^i  (fig.  7)  be  two  objects  and  a.a.^,  h.fi^  their  images,  A^B^  the 
distance    of  the    objects,    AM.   that    of  the    images,    ^i^   the    index  of  refraction  of 


ON    THE    GENERAL     LAWS    OF    OPTICAL    INSTRUMENTS.  281 

the    medium    in    which   the   objects   are,    /a,   that    of    the    medium   in    which   tlie 
images  are,  then 

«,a,  X  /^,y8,  _      a,a,  x  h.fi., 
^'     A  A     ~^'     A,B.,    '' 

approximately,  when  the  objects  are  small. 

Since  a,  is  the  image  of  a^,  the  reduced  path  of  the  ray  a,6,a,,  will  be 
equal  to  that  of  a^^a„_,  and  the  reduced  paths  of  the  rays  a^/3,cu  and  a,/Aa,  will 
be  equal. 

Also  because  l)^^  and  h.^„  are  conjugate  foci,  the  reduced  paths  of  the 
rays  b^ajj,  and  h^aj),,  and  of  ^ia,,/8j  and  ^,a.,/3,  will  be  equal.  So  that  the 
reduced  paths 

afi,  +  h,a^  =  a^ySj  +  ^.a^ 

aJ3,  +  I3,0L,  =  tti^i  +  b.cL, 

feiOj  +  Oj^j  =  b^a^  +  alt., 

these   being   still   the   reduced  paths   of  the   rays,    that   is,    the    length     of    each 
ray  multiplied  by  the  index  of  refraction  of  the  medium. 

If  the  figure  is  symmetrical  about  the  axis,   we  may  write  the  equation 

Fi  (aA  -  «i^i)  =  /^2  (aA  -  ci-A), 
where  aJS^,   &c.  are  now  the  ax^tual  lengths  of  the  rays  so  named. 


Now  aA'  =  A,B;'  + 1  (a,a,  +  b^.f, 


so  that  a^i  —  aj)^  =  OiC^  x  6^8, , 

a.a,  X  61)8, 


and  ft,  (a^  -  aj),)  =  fi^ 


a  A  +  aj)^ 


Similarly  /x,  (a^  -  a,&,)  =  fi,  ^^^^^j^' 

So  that  the  equation  /x,  ^  ,    "T'  =  /x^  — ^— — ,   , 


282  ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS. 

is  true  accurately,  and  since  when  the  objects  are  small,  the  denominators  are 
nearly  2A,B^  and  2A^„  the  proposition  is  proved  approximately  true. 

Using  the  expressions  of  Prop.  III.,  this  equation  becomes 

1  xy 

Now   by  Prop.  III.,   when   x  and  y  are  different,  the  focal  lengths  /,  and  /, 

are 

.  xy  ^  1 

^1      'x-y        ^        y  —  ^ 

therefore  -^  =  -^  =  -    by  the  present  theorem. 

So  that  in  any  instrument,  not  a  telescope,  the  focal  lengths  are  directly  as 
the  indices  of  refraction  of  the  media  to  which  they  belong.  If,  as  in  most 
cases,  these  media  are  the  same,  then  the  two  focal  distances  are  eqiial 

When  x  =  y,   the  instrument  becomes  a  telescope,  and  we  have,  by  Prop.  V., 

l  =  x    and  n=-;    and  therefore  by  this  theorem 

m     n' 

We  may  find  I  experimentally  by  measuring  the  actual  diameter  of  the 
image  of  a  known  near  object,  such  as  the  aperture  of  the  object  glass.  If  0  be 
the  diameter  of  the  aperture  and  o  that  of  the  circle  of  light  at  the  eye-hole 
(which  is  its  image),  then 

From  this  we  find  the  elongation  and  the  angular  magnifying  power 

n  =  ^'l\   and   m  =  ^'y. 

When  ix,  =  fi„  as  in  ordinary  cases,  m  =  y  =  -,  which  is  Gauss'  rule  for  deter- 
mining the  magnifying  power  of  a  telescope. 


ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS.  283 

Prop.  IX.  It  is  impossible,  bj  means  of  any  combination  of  reflexions 
and  refractions,  to  produce  a  perfect  image  of  an  object  at  two  different  distances, 
unless  the  instrument  be  a  telescope,  and 

l  =  n=-,        m=l. 

It  appears  from  the  investigation  of  Prop.  VIII.  that  the  results  there 
obtained,  if  true  when  the  objects  are  very  small,  will  be  incorrect  when  the 
objects  are  large,   unless 

ajSi  +  tti^i  :  a^^  +  a,h  ::  A^B^  :  A^^, 

and  it  is  easy  to  prove  that  this  cannot  be,  unless  all  the  Hnes  in  the  one  figure 
are  proportional  to  the  corresponding  lines  in  the  other. 

In  this  way  we  might  show  that  we  cannot  in  general  have  an  astigmatic, 
plane,  undistorted  image  of  a  plane  object.  But  we  can  prove  that  we  cannot 
get  perfectly  focussed  images  of  an  object  in  two  positions,  even  at  the  expense 
of  curvature  and  distortion. 

We  shall  first  prove  that  if  two  objects  have  perfect  images,  the  reduced 
path  of  the  ray  joining  any  given  points  of  the  two  objects  is  equal  to  that 
of  the   ray  joining  the  corresponding  points  of  the  images. 

Let   tto  (fig.    8)  be  the  perfect  image  of  a^  and  yS^  of  /B^.     Let 

Ajai  =  a^,   BJ3,  =  b„   Ajx^  =  a^,    B.J3.,  =  b.,   A^B^  =  c^,   A^^  =  c^. 

Draw  a^D^  parallel   to   the   axis   to    meet   the   plane   B^y   and   aJD,  to   the   plane 
of  A. 

Since  everything  is  symmetrical  about  the  axis  of  the  instrument  we  shall 
have  the  angles  D^Bfi^  =  D.M.fi,  =  d,  then  in  either  figure,  omitting  the  sufl&xes, 

=  c'  +  a'  +  b'-2ahcose. 

It  has  been  shown  in  Prop.  VIII.  that  the  difference  of  the  reduced  paths 
of  the  rays  aj)^,  afi^  in  the  object  must  be  equal  to  the  difference  of  the  reduced 
paths  of  a^^j,  a^^  in  the  image.     Therefore,  since  we  may  assume  any  value  for  6 

/^i  J{(^x  +  &i'  +  Ci*  -  lajb,  cos  6)  -  fi,  J{a^  +  h^  +  c^  -  2a,h  cos  6) 


284  ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS. 

13  constant  for  all  values  of  6.     This  can  be  only  when 

and  fi,  J{aJ),)  =fi,J  (aM,), 

which  shows  that  the  constant  must  vanish,  and  that  the  lengths  of  lines 
joining  corresponding  points  of  the  objects  and  of  the  images  must  be  inversely 
as  the  indices  of  refraction  before  incidence  and  after  emergence. 

Next  let  ABC,  DEF  (fig.  9)  represent  three  points  in  the  one  object 
and  three  points  in  the  other  object,  the  figure  being  drawn  to  a  scale  so  that 
all  the  lines  in  the  figure  are  the  actual  lines  multiplied  by  /Xj.  The  lines  of 
the  figure  represent  the  reduced  paths  of  the  rays  between  the  corresponding 
points  of  the  objects. 

Now  it  may  be  shown  that  the  form  of  this  figure  cannot  be  altered  with- 
out altering  the  length  of  one  or  more  of  the  nine  lines  joining  the  points  ABC 
to  DEF.  Therefore  since  the  reduced  paths  of  the  rays  in  the  image  are  equal 
to  those  in  the  object,  the  figure  must  represent  the  image  on  a  scale  of  /n, 
to  1,  and  therefore  the  instrument  must  magnify  every  part  of  the  object  alike 
and  elongate  the  distances  parallel  to  the  axis  in  the  same  proportion.  It  is 
therefore  a  telescope,  and  m=l. 

If  iJi,  =  ix,,  the  image  is  exactly  equal  to  the  object,  which  is  the  case  in 
reflexion  in  a  plane  mirror,  which  we  know  to  be  a  perfect  instrument  for  all 
distances. 

The  only  case  in  which  by  refraction  at  a  single  surface  we  can  get  a 
perfect  image  of  more  than  one  point  of  the  object,  is  when  the  refracting 
surface  is  a  sphere,  radius  r,  index  /x,   and  when   the   two   objects    are    spherical 

surfaces,  concentric  with  the  sphere,  their  radii  being    - ,   and   r ;    and    the    two 

images  also  concentric  spheres,  radii  /ar,  and  r. 

In  this  latter  case  the  image  is  perfect,  only  at  these  particular  distances 
and  not  generally. 

I  am  not  aware  of  any  other  case  in  which  a  perfect  image  of  an  object 
can  be  formed,  the  rays  being  straight  before  they  enter,  and  after  they  emerge 
from  the  instrument.  The  only  case  in  which  perfect  astigmatism  for  all  pencils 
has    hitherto   been   proved   to   exist,   was   suggested   to   me   by   the   consideration 


VOL.  I.  PLATE  IV. 


Ftg.l 


Fisr.2 


h 

^^       c 

H 

I 

Q 

1^3 

a. 

--^^ 

/ 

1 

^.     ^ 

?,       A 

^                    ^.^ 

^F,                B,     6i       A 

/ 

^.           / 

, 

;                «% 

^>^^% 

9, 

-^^^ 

Fx.y.3 

|-\ 

c 

7                     P           P"""-^^ 

— ^-^                  6 

i                ^ 

^'5'-^      C 


ba 

«^ 

^a^^ 

1 

5 

t 

a 

^ 

1 

a. 

^ 

^^^^^ 

nr 

L 

^. 

A 

2 

} 

0 

2 

/ 

^  I 

h 

A 

> 

0 

1 

Fig.S 


vol.  I.  PLATE  IV. 


Fvcf.6 


9t^ 

92 

,        hF^    '^i 

n 

"i     ^^ 

1 

1 

^. 

A        9i 

Ftsf.  7 


F19  S 


■ 

h 

^2^—^——-' — ' — ' 

^1 

«i 

^ 

\. — 

^__,^^^- 

f' 

^ 

\ 

V-^ 

-^^ 

H — — 

• , 

' 

A^ 

^^H^ 

^"-^^ — ^^ 

^r"^ 

^ 

__  . 

v^ 

^ 

\^ 

'A 

»i 

^1 

'. 

^2 

/ 

*'J 

^1 

\  1 

/ 

^    V 

J 

vy 

i 

Fig.  9        E^ 


ON    THE    GENERAL    LAWS    OF    OPTICAL    INSTRUMENTS.  285 

of  the  structure  of  the  crystalline  lens  in  fish,  and  was  published  in  one  of 
the  problem-papers  of  the  Camhiidge  and  Dublin  Mathematical  Journal.  My 
own  method  of  treating  that  problem  is  to  be  found  in  that  Journal,  for 
February,  1854.  The  case  is  that  of  a  medium  whose  index  of  refraction  varies 
with  the  distance  from  a  centre,  so  that  if  fi,  be  its  value  at  the  centre,  a 
a  given  line,  and  r  the  distance  of  any  point  where  the  index  is  /x,  then 


/^  =  /Ao 


a'  +  r'' 


The  path  of  every  ray  within  this  medium  is  a  circle  in  a  plane  passing  through 
the  centre  of  the  medium. 

Every    ray    from   a   point   in   the    medium,    distant   b   from   the   centre,    will 

converge  to  a  point  on  the  opposite  side  of  the  centre  and  distant  from  it  ^  . 

It  will  be  observed  that  both  the  object  and  the  image  are  included  in 
the  variable  medium,  otherwise  the  images  would  not  be  perfect.  This  case 
therefore  forms  no  exception  to  the  result  of  Prop.  IX.,  in  which  the  object  and 
image  are  supposed  to  be  outside  tho  instrument. 

Aberdeen,   12th  Jan.,   1858. 


[From  the  Proceedings  of  the  Royal  Society  of  Edinburgh,  Vol.  rv.] 


XYIII.      On    Theories    of  the    Constitution   of  Saturn's   Rings. 

The  planet  Saturn  is  surrounded  by  several  concentric  flattened  rings,  which 
appear  to  be  quite  free  from  any  connection  with  each  other,  or  with  the  planet, 
except  that  due  to  gravitation. 

The  exterior  diameter  of  the  whole  system  of  rings  is  estimated  at  about 
176,000  miles,  the  breadth  from  outer  to  inner  edge  of  the  entire  system, 
36,000   miles,   and   the   thickness  not   more   than    100   miles. 

It   is   evident    that    a    system    of   this    kind,   so    broad    and    so    thin,    must 
depend   for   its  stability  upon  the  dynamical  equihbrium  between  the  motions  of 
each   part   of  the   system,    and  the   attractions   which    act    on    it,   and    that    the 
cohesion   of  the   parts   of  so   large   a   body   can   have   no   effect   whatever   on  its 
motions,  though   it   were   made   of  the   most   rigid   material  known  on  earth.     It 
is   therefore   necessary,   in   order  to   satisfy   the   demands   of   physical    astronomy, 
to   explain   how  a  material   system,  presenting  the  appearance  of  Saturn's  Kings, 
can  be  maintained  in  permanent  motion  consistently  with  the  laws  of  gravitation. 
The  principal  hypotheses  which  present  themselves  are  these — 
I.     The  rings  are  solid  bodies,  regular  or  irregular. 
II.     The  rings  are  fluid  bodies,  liquid  or  gaseous. 
in.     The  rings  are  composed  of  loose  materials. 

The  results  of  mathematical  investigation  appHed  to  the  first  case  are, — 

1st.     That  a  uniform  ring  cannot  have  a  permanent  motion. 

2nd.  That  it  is  possible,  by  loading  one  side  of  the  ring,  to  produce 
stability  of  motion,  but  that  this  loading  must  be  very  great  compared  with 
the  whole  mass  of  the  rest  of  the  ring,  being  as  82  to  18. 


ON    THEORIES    OF    THE    CONSTITUTION    OF    SATURN's    RINGS,  287 

3rd.  That  this  loading  must  not  only  be  very  great,  but  very  nicely 
adjusted;  because,  if  it  were  less  than  '81,  or  more  than  83  of  the  whole, 
the  motion  would  be  unstable. 

The  mode  in  which  such  a  system  would  be  destroyed  would  be  by  the 
collision  between  the  planet  and  the  inside  of  the  ring. 

And  it  is  evident  that  as  no  loading  so  enormous  in  comparison  with  the 
ring  actually  exists,  we  are  forced  to  consider  the  rings  as  fluid,  or  at  least 
not  solid ;  and  we  find  that,  in  the  case  of  a  fluid  ring,  waves  would  be  gene- 
rated, which  would  break  it  up  into  portions,  the  number  of  which  would 
depend  on  the  mass  of  Saturn  directly,  and  on  that  of  the  ring  inversely. 

It  appears,  therefore,  that  the  only  constitution  possible  for  such  a  ring  is 
a  series  of  disconnected  masses,  which  may  be  fluid  or  solid,  and  need  not  be 
equal.  The  \iomplicated  internal  motions  of  such  a  ring  have  been  investigated, 
and  found  to  consist  of  four  series  of  waves,  which,  when  combined  together, 
will  reproduce  any  form  of  original  disturbance  with  all  its  consequences.  The 
motion  of  one  of  these  waves  was  exhibited  to  the  Society  by  means  of  a  small 
mechanical  model  made  by  Ramage  of  Aberdeen. 

This  theory  of  the  rings,  being  indicated  by  the  mechanical  theory  as  the 
only  one  consistent  with  permanent  motion,  is  further  confirmed  by  recent  obser- 
vations on  the  inner  obscure  ring  of  Saturn.  The  limb  of  the  planet  is  seen 
through  the  substance  of  this  ring,  not  refracted,  as  it  would  be  through  a 
gas  or  fluid,  but  in  its  true  position,  as  would  be  the  case  if  the  light  passed 
through  interstices  between  the  separate  particles  composing  the  ring. 

As  the  whole  investigations  are  shortly  to  be  published  in  a  separate  form, 
the  mathematical  methods  employed  were  not  laid  before  the  Society. 


XIX.     On  the  Stability  of  the  motion  of  Saturn's  Rings. 


[An  Essay,  which  obtained  the  Adams  Prize  for  the  year  1856,  in  the  University 

of  Cambridge.] 


ADVERTISEMENT. 

The  Subject  of  the  Prize  was  announced  in  the  following  terms ; — 

The  University  having  accepted  a  fimd,  raised  by  several  members  of  St  John's  Collegp, 
for  the  purpose  of  founding  a  Prize  to  be  called  the  Adams  Prize,  for  the  best  Essay 
on  some  subject  of  Pure  Mathematics,  Astronomy,  or  other  branch  of  Natural  Pliilosophy, 
the  Prize  to  be  given  once  in  two  years,  and  to  be  open  to  tlhe  competition  of  all  persons 
who  have  at  any  time  been  admitted  to  a  degree  in  this   University: — 

The  Examiners  give  Notice,  that  the  following  is  the  subject  for  the  Prize  to  be  adjudged 
in  1857:— 

The  Motions  of  iSaturn's  Rings. 

***  The  problem  may  be  treated  on  the  supposition  that  the  system  of  Rings  is  exactly  or 
very  approximately  concentric  with  Saturn  and  symmetrically  disposed  about  the  plane  of  his  Equator, 
and  different  hypotheses  may  be  made  respecting  the  physical  constitution  of  the  Rings.  It  may 
be  supposed  (1)  that  they  are  rigid:  (2)  that  they  ai-e  fluid,  or  in  part  aeriform:  (3)  that  they 
consist  of  masses  of  matter  not  mutually  coherent.  The  question  will  be  considered  to  be  answered 
by  ascertaining  on  tliese  hypotheses  severally,  whether  the  conditions  of  mechanical  stability  are 
satisfied  by  the  mutual  attractions  and  motions  of  the  Planet  and  the  Rings. 

It  is  desirable  that  an  attempt  should  also  be  made  to  determine  on  which  of  the  above 
hypotheses  the  appearances  both  of  the  bright  Rings  and  the  recently  discovered  dark  Ring  may 
be  most  satisfactorily  explained;  and  to  indicate  any  causes  to  which  a  change  of  form,  such  as 
is  supposed  from  a  comparison  of  modern  with  the  earlier  observations  to  have  taken  place,  may 
be  attributed. 

E.  GUEST,   rice-Chancellor. 

J.  CHALLIS. 

S.  PARKINSON. 

W.  THOMSON. 
March  23,   1855. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's     RINGS.  289 


CONTENTS. 

Nature  of  the  Problem 290 

Laplace's  investigaticm  of  the  Equilibrium  of  a  Ring,  otuI  its  minimum  density 29i 

Hit  proof  that  Oie  platie  of  tlie  Rings  will  follow  that  of  Saturn's  Equator — that  a  solid  uniform  Ring  is 

necessarily  unstable            293 

Further  investigation  required— Theory  of  an  Irregular  Solid  Ring  leads  to  the  result  that  to  ensure  stability 

the  irregularity  mitst  be  enormous 294 

Theory  of  a  fluid  or  discontinuous  Ring  resolves  itself  into  the  investigation  of  a  series  of  waves          .        .  295 

PART  I. 

ON   THE   MOTION   OF   A   RIGID   BODY   OF   ANY   FORM   ABOUT 
A   SPHERE. 

Equations  of  Motion 296 

Problem  I.     To  find  the  conditions  under  which  a  uniform  motion  of  the  Ring  is  possible  .        .        .         298 

Problem  II.     To  find  the  equations  of  the  motion  when  slightly  distxirbed 299 

Problem  III.     To  reduce  the  three  siinultaneou^  equations  of  motion  to  the  form  of  a  single  linear  equation        300 
Problem  IV.     To  determijie  whether  the  motion  of  the  Ring  is  stable  or  unstable,  by  meayis  of  the  relations 

of  the  coefficients  A,  B,  C 301 

Problem  V.     To  find  the  centre  of  gravity,  the  radius  of  gyration,  and  the  variations  of  the  potential  Tieaf 

the  centre  of  a  circular  ring  of  small  but  variable  section 302 

Problem  VI.     To  determine  the  condition  of  stability  of  the  motion  in  terms  of  the  coeffilcierits  f,  g,  h,  which 

indicate  the  distribution  of  mass  in  the  ring 306 

RB6ULTS.  I«^,  a  uniform  ring  is  unstable.  2nd,  a  ring  varying  in  section  according  to  the  law  of  sines  is 
unstable.  3rd,  a  uniform  ring  loaded  with  a  heavy  particle  may  be  stable,  provided  the  mass  of  the 
particle  be  between  'SlSSeS  and  "8279  of  the  whole.  Case  in  which  the  ring  is  to  the  particle  as  18 
«o  82 307 

PART    II. 

ON  THE  MOTION   OF  A   RING,   THE  PARTS  OF  WHICH   ARE  NOT 
RIGIDLY   CONNECTED. 

1.  General  Statemeiii  of  the  Problem,  and  limitation  to  a  nearly  uniform  ring 310 

2.  Notation 311 

3.  Expansion  of  a  function  in  terms  of  sines  and  cosines  of  multiples  of  the  variable       .        .        .         .  311 

4.  Magnitude  and  direction  of  attraction  between  two  elements  of  a  disturbed  Ring 312 

5.  Resultant  attractions  on  any  one  of  a  ring  of  equal  satellites  disturbed  in  any  way      ....  313 
Note.     Calculated  values  of  these  attractions  in  particular  cases      .        .        .        .        .        .         .         .  314 

6.  Equations  of  motion  of  a  satellite  of  the  Ring,  and  biquadratic  equation  to  determine  the  wave-velocity  31.5 

7.  A  ring  of  satellites  may  always  be  rendered  stable  by  increasing  the  mass  of  the  central  body      .  .  317 

8.  Relation  between  the  number  and  mass  of  satellites  and  the  mass  of  the  central  body  necessary  to 

ensure  stability.     S>-4352,x2R 318 

9.  Solution  of  the  biquadratic  equation  when  the  mass  of  t/ie  Ring  is  small ;  and  complete  e.rprcssions 

for  the  Tnotion  of  each  satellite  ..............         319 

10.     Each  satellite  moves  {relatively  to  tJie  ring)  in  an  ellipse 321 


290  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

11.  Each  satellite  moves  absolutely/   in  S2>ace  in  a  curve  which  is  nearly  an  ellipse  for  the  large  values 

of  n,  and  a  spiral  of  many  nearly  circular  coih  when  n  is  small 321 

12.  The  form  of  the  ring  at  a  given  instant  is  a  series  of  undulations 322 

13.  These  uiidvlations  travel  round  the  ring  with  velocity relative  to  the  ring,  and  a absolutely  323 

14.  General  Solution  of  the  Problem — Given  the  position  and  motion  of  every  satellite  at  any  one  time, 
to  calculate  the  position  and  motion  of  any  satellite  at  any  other  time,  provided  that  the  condition 

of  stability  is  fulfilled 323 

15.  Calculation  of  the  effect  of  a  periodic  external  disturbing  force 326 

16.  Treatment  of  disturbing  forces  in  general 328 

17.  Theory  of  free  waves  and  forced  waves 329 

18.  Motion  of  the  ring  when  the  conditions  of  stability  are  net  fulfilled.      Two  different  ways  in  which 

the  ring  may  be  broken  up 330 

19.  Motion  of  a  riyig  of  unequal  satellites 335 

20.  Motion  of  a  ring  composed  of  a  clowi  of  scattered  particles 336 

21.  Calculation  of  the  forces  arising  from  the  displacements  of  such   a  system 337 

22.  Application  to  the  case  of  a  ring  of  this  kind.      The  mean  density  must  be  excessively  s^nall,  which 

is  iTiconsistent  with  its  moving  as  a  whole      ............  338 

23.  On    the  forces   arising  from    inequalities    in  a    thin    stratum  of  gravitating  incompressible  fluid  of 
indefinite  extent 338 

24.  Application  to  the  case  of  a  flattened  fluid  ring,  moving  with  uniform  angular  velocity.     Such  a 

ring  will  be  broken  up  into  portions  which  may  continue  to  revolve  as  a  ring  of  satellites   .        .        .  344 

ON  THE  MUTUAL  PERTUKBATIONS  OF  TWO  RINGS. 

25.  Application  of  the  general  theory  of  free  and  forced  waves 345 

26.  To  determine  the  attractions  between  the  rings 346 

27.  To  form  the  equations  of  motion 348 

28.  Method  of  determining  the  reaction  of  the  forced  wave  on  tlve  free  wave  which  produced  it         .        .  349 

29.  Cases  in  which  the  perturbations  increase  indefinitely      .        .         ........  351 

30.  Application  to  the  theory  of  an  indefinite  number  of  concentric  rings 352 

31.  On  the  effect  of  long-continued  disturbances  on  a  system  of  rings 352 

32.  On  the  effect  of  collisions  among  the  parts  of  a  revolving  system           .......  354 

33.  On  the  effect  of  internal  friction  in  a  fluid  ring              354 

Ilecapitulation    of  the    Theory  of  the  Motion  of  a  Rigid  Ring.    Reasons  for  ryecting  tlie  hypothesis  of 

rigidity 356 

Recapitulation  of  the   Theory  of  a   Ring  of  Equ/il  Satellites 360 

Description  of  a  working  model  shewing  the  motions  of  such  a  system 363 

Theory  of  Rings  of  various  constitutions 367 

Mutual  action  of  Two  Rings 370 

Case  of  many  concentric  Rings,  &c. .        .  371 

General  Conclusions 372 

Appendix.     Extract  of  a  letter  from  Professor  W.  Thomson,  of  Glasgow,  giving  a  solution  of  t/ie  Pro- 
blem of  a  Rigid  Ring 374 


There  are  some  questions  in  Astronomy,  to  which  we  are  attracted  rather 
on  account  of  their  pecuHarity,  as  the  possible  illustration  of  some  unknown 
principle,  than   from   any  direct   advantage   which   their   solution   would   afford   to 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's     RINGS.  291 

mankind.  The  theory  of  the  Moon's  inequalities,  though  in  its  first  stages  it 
presents  theorems  interesting  to  all  students  of  mechanics,  has  been  pursued  into 
such  intricacies  of  calculation  as  can  be  followed  up  only  by  those  who  make 
the  improvement  of  the  Lunar  Tables  the  object  of  their  lives.  The  value  of 
the  labours  of  these  men  is  recognised  by  all  who  are  aware  of  the  importance 
of  such  tables  in  Practical  Astronomy  and  Navigation.  The  methods  by  which 
the  results  are  obtained  are  admitted  to  be  sound,  and  we  leave  to  professional 
astronomers  the  labour  and  the  merit  of  developing  them. 

The  questions  which  are  suggested  by  the  appearance  of  Saturn's  Rings 
cannot,  in  the  present  state  of  Astronomy,  call  forth  so  great  an  amount  of 
labour  among  mathematicians.  I  am  not  aware  that  any  practical  use  has  been 
made  of  Saturn's  Rings,  either  in  Astronomy  or  in  Navigation.  They  are  too 
distant,  and  too  insignificant  in  mass,  to  produce  any  appreciable  effect  on  the 
motion  of  other  parts  of  the  Solar  system;  and  for  this  very  reason  it  is  diflS- 
cult  to  determine  those  elements  of  their  motion  which  we  obtain  so  accurately 
in  the  case  of  bodies  of  greater  mechanical  importance. 

But  when  we  contemplate  the  Rings  from  a  purely  scientific  point  of  view, 
they  become  the  most  remarkable  bodies  in  the  heavens,  except,  perhaps,  those 
still  less  useful  bodies — the  spiral  nebulae.  When  we  have  actually  seen  that 
great  arch  swung  over  the  equator  of  the  planet  without  any  visible  connexion, 
we  cannot  bring  our  minds  to  rest.  We  cannot  simply  admit  that  such  is  the 
case,  and  describe  it  as  one  of  the  observed  facts  in  nature,  not  admitting  or 
requiring  explanation.  We  must  either  explain  its  motion  on  the  principles  of 
mechanics,  or  admit  that,  in  the  Saturnian  realms,  there  can  be  motion  regu- 
lated by  laws  which  we  are  unable  to  explain. 

The  arrangement  of  the  rings  is  represented  in  the  figure  (l)  on  a  scale 
of  one  inch  to  a  hundred  thousand  miles.  S  is  a  section  of  Saturn  through 
his  equator,  A,  B  and  C  are  the  three  rings.  A  and  B  have  been  known  for 
200  years.  They  were  mistaken  by  Galileo  for  protuberances  on  the  planet  itself, 
or  perhaps  satellites.  Huyghens  discovered  that  what  he  saw  was  a  thin  flat 
ring  not  touching  the  planet,  and  Ball  discovered  the  division  between  A  and  B. 
Other  divisions  have  been  observed  splitting  these  again  into  concentric  rings, 
but  these  have  not  continued  visible,  the  only  well-established  division  being  one 
in  the  middle  of  A.  The  third  ring  C  was  first  detected  by  Mr  Bond,  at 
Cambridge  U.S.  on  November  15,  1850;  Mr  Dawes,  not  aware  of  Mr  Bond's 
discovery,  observed   it   on    November    29th,  and   Mr  Lassel  a  few  days  later.     It 


292  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

gives  little  light  compared  with  the  other  rings,  and  is  seen  where  it  crosses 
the  planet  as  an  obscure  belt,  but  it  is  so  transparent  that  the  limb  of  the 
planet  is  visible  through  it,  and  this  without  distortion,  shewing  that  the  rays 
of  light  have  not  passed  through  a  transparent  substance,  but  between  the 
scattered  particles  of  a  discontinuous  stream. 

It  is  difficult  to  estimate  the  thickness  of  the  system ;  according  to  the 
best  estimates  it  is  not  more  than  100  miles,  the  diameter  of  A  being  176,418 
miles;  so  that  on  the  scale  of  our  figure  the  thickness  would  be  one  thousandth 
of  an  inch. 

Such  is  the  scale  on  which  this  magnificent  system  of  concentric  rings  is 
constructed;  we  have  next  to  account  for  their  continued  existence,  and  to 
reconcile  it  with  the  known  laws  of  motion  and  gravitation,  so  that  by  rejecting 
every  hypothesis  which  leads  to  conclusions  at  variance  with  the  facts,  we  may 
learn  more  of  the  nature  of  these  distant  bodies  than  the  telescope  can  yet 
ascertain.  We  must  account  for  the  rings  remaining  suspended  above  the  planet, 
concentric  with  Saturn  and  in  his  equatoreal  plane ;  for  the  flattened  figure  of  the 
section  of  each  ring,  for  the  transparency  of  the  inner  ring,  and  for  the  gradual 
approach  of  the  inner  edge  of  the  ring  to  the  body  of  Saturn  as  deduced 
from  all  the  recorded  observations  by  M.  Otto  Struvd  {Sur  les  dimensions  des 
Anneaux  de  Saturne — Recueil  de  Memoires  Astronomiques,  Poulkowa,  15  Nov. 
1851).  For  an  account  of  the  general  appearance  of  the  rings  as  seen  from  the 
planet,  see  Lardner  on  the  Uranography  of  Saturn,  Mem.  of  the  Astronomical 
Society,  1853.  See  also  the  article  "Saturn"  in  Nichol's  Cyclopcedia  of  the 
Physical  Sciences. 

Our  curiosity  with  respect  to  these  questions  is  rather  stimulated  than 
appeased  by  the  investigations  of  Laplace.  That  great  mathematician,  though 
occupied  with  many  questions  which  more  imperiously  demanded  his  attention, 
has  devoted  several  chapters  in  various  parts  of  his  great  work,  to  points  con- 
nected with  the  Saturnian  System. 

He  has  investigated  the  law  of  attraction  of  a  ring  of  small  section  on  a 
point  very  near  it  {Mec.  Cel.  Liv.  iii.  Chap,  vi.),  and  from  this  he  deduces  the 
equation   from  which   the   ratio   of  the   breadth  to  the  thickness  of  each  ring  is 

to  be  found, 

E'  p  X(X-l) 

^~3a'p      (\+l)  (3X^+1)' 

where   R   is  the  radius  of  Saturn,  and  p  his  density;   a  the  radius  of  the  ring, 


ON    THE    STABIUTY    OF    THE    MOTION    OF    SATURN's    RINGS.  293 

and  p  its  density;  and  X  the  ratio  of  the  breadth  of  the  ring  to  its  thick- 
ness. The  equation  for  determining  X  when  e  is  given  has  one  negative  root 
which  must  be  rejected,  and  two  roots  which  are  positive  while  e<0"0543,  and 
impossible  when  e  has  a  greater  value.  At  the  critical  value  of  e,  X  =  2-594 
nearly. 

The  fact  that  X  is  impossible  when  e  is  above  this  value,  shews  that  the 
ring  cannot  hold  together  if  the  ratio  of  the  density  of  the  planet  to  that  of 
the  ring  exceeds  a  certain  value.  This  value  is  estimated  by  Laplace  at  I'S, 
assuming  a  =  2R. 

We  may  easily  follow  the  physical  interpretation  of  this  result,  if  we  observe 
that  the  forces  which  act  on  the  ring  may  be  reduced  to — 

(1)  The  attraction  of  Saturn,  varying  inversely  as  the  square  of  the  dis- 
tance from  his  centre. 

(2)  The  centrifugal  force  of  the  particles  of  the  ring,  acting  outwards,  and 
varying  directly  as  the  distance  from  Saturn's  polar  axis. 

(3)  The  attraction  of  the  ring  itself,  depending  on  its  form  and  density, 
and  directed,  roughly  speaking,  towards  the  centre  of  its  section. 

The  first  of  these  forces  must  balance  the  second  somewhere  near  the  mea,n 
distance  of  the  ring.  Beyond  this  distance  their  resultant  will  be  outwards, 
within  this  distance  it  will  act  inwards. 

If  the  attraction  of  the  ring  itself  is  not  sufl&cient  to  balance  these  residual 
forces,  the  outer  and  inner  portions  of  the  ring  will  tend  to  separate,  and  the 
ring  will  be  split  up ;  and  it  appears  from  Laplace's  result  that  this  will  be 
the  case  if  the  density  of  the  ring  is  less  than  ^  of  that  of  the  planet. 

This  condition  applies  to  all  rings  whether  broad  or  narrow,  of  which  the 
parts  are  separable,  and  of  which  the  outer  and  inner  parts  revolve  with  the 
same  angular  velocity. 

Laplace  has  also  shewn  (Li v.  v.  Chap,  iii.),  that  on  account  of  the  oblate- 
ness  of  the  figure  of  Saturn,  the  planes  of  the  rings  will  follow  that  of  Saturn's 
equator  through  every  change  of  its  position  due  to  the  disturbing  action  of 
other  heavenly  bodies. 

Besides  this,  he  proves  most  distinctly  (Liv.  iii.  Chap,  vi.),  that  a  solid  uni- 
form ring  cannot  possibly  revolve  about  a  central  body  in  a  permanent  manner, 
for  the  slightest  displacement  of  the  centre  of  the  ring  from  the  centre  of  the 
planet    would    originate   a   motion   which    would    never    be    checked,    and    would 


294  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

inevitably  precipitate  the  ring  upon  the  planet,  not  necessarily  by  breaking  the 
ring,  but  by  the  inside  of  the  ring  falling  on  the  equator  of  the  planet. 

He  therefore  infers  that  the  rings  are  irregular  solids,  whose  centres  of 
gravity  do  not  coincide  with  their  centres  of  figure.  We  may  draw  the  con- 
clusion more  formally  as  follows,  "If  the  rings  were  solid  and  uniform,  their 
motion  would  be  unstable,  and  they  would  be  destroyed.  But  they  are  not 
destroyed,  and  their  motion  is  stable;  therefore  they  are  either  not  uniform  or 
not  solid." 

I  have  not  discovered""  either  in  the  works  of  Laplace  or  in  those  of  more 
recent  mathematicians,  any  investigation  of  the  motion  of  a  ring  either  not  uni- 
form or  not  solid.  So  that  in  the  present  state  of  mechanical  science,  we  do 
not  know  whether  an  irregular  solid  ring,  or  a  fluid  or  disconnected  ring,  can 
revolve  permanently  about  a  central  body;  and  the  Saturnian  system  still  re- 
mains an  unregarded  witness  in  heaven  to  some  necessary,  but  as  yet  unknown, 
development  of  the  laws  of  the  universe. 

We  know,  since  it  has  been  demonstrated  by  Laplace,  that  a  uniform  solid 
ring  cannot  revolve  permanently  about  a  planet.  We  propose  in  this  Essay  to 
determine  the  amount  and  nature  of  the  irregularity  which  would  be  required 
to  make  a  permanent  rotation  possible.  We  shall  find  that  the  stability  of  the 
motion   of  the   ring   would   be   ensured  by  loading  the  ring  at  one  point  with  a 

*  Since  this  -was  written,  Prof.  Challis  has  pointed  out  to  me  three  important  papers  in  Gould's 
Astronomical  Journal: — Mr  G.  P.  Bond  on  the  Rings  of  Saturn  (May  1851)  and  Prof.  B.  Pierce  of 
Harvard  University  on  the  Constitution  of  Saturn's  Rings  (June  1851),  and  on  the  Adams'  Prize 
Problem  for  1856  (Sept.  1855).  These  American  mathematicians  have  both  considered  the  conditions 
of  statical  equilibrium  of  a  transverse  section  of  a  ring,  and  have  come  to  the  conclusion  that  the 
rings,  if  they  move  each  as  a  whole,  must  be  very  narrow  compared  with  the  observed  rings,  so 
that  in  reality  there  must  be  a  great  number  of  them,  each  revolving  with  its  own  velocity.  They 
have  also  entered  on  the  question  of  the  fluidity  of  the  rings,  and  Prof.  Pierce  has  made  an 
investigation  as  to  the  permanence  of  the  motion  of  an  irregular  solid  ring  and  of  a  fluid  ring. 
The  paper  in  which  these  questions  are  treated  at  large  has  not  (so  far  as  I  am  aware)  been 
pxiblished,  and  the  references  to  it  in  Gould's  Journal  are  intended  to  give  rather  a  popular  account 
of  the  results,  than  an  accurate  outline  of  the  methods  employed.  In  treating  of  the  attractions  of 
an  irregular  ring,  he  makes  admirable  use  of  the  theory  of  potentials,  but  his  published  investi- 
gation of  the  motion  of  such  a  body  contains  some  oversights  which  are  due  perhaps  rather  to  the 
imperfections  of  popular  language  than  to  any  thing  in  the  mathematical  theory.  The  only  part  of 
the  theory  of  a  fluid  ring  which  he  has  yet  given  an  account  of,  is  that  in  which  he  considers 
the  form  of  the  ring  at  any  instant  as  an  ellipse;  corresponding  to  the  case  where  n  =  u),  and 
m=l.  As  I  had  only  a  limited  time  for  reading  these  papers,  and  as  I  could  not  ascertain  the 
methods  used  in  the  original  investigations,  I  am  unable  at  present  to  state  how  far  the  results  of 
this  essay  agree  with  or  differ  from  those  obtained  by  Prof.  Pierce. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  295 

heavy  satellite  about  4-^  times  the  weight  of  the  ring,  but  this  load,  besides 
being  inconsistent  with  the  observed  appearance  of  the  rings,  must  be  far  too 
artificially  adjusted  to  agree  with  the  natural  arrangements  observed  elsewhere, 
for  a  very  small  error  in  excess  or  defect  would  render  the  ring  again  unstable. 

We  are  therefore  constrained  to  abandon  the  theory  of  a  solid  ring,  and 
to  consider  the  case  of  a  ring,  the  parts  of  which  are  not  rigidly  connected, 
as  in  the  case  of  a  ring  of  independent  satellites,  or  a  fluid  ring. 

There  is  now  no  danger  of  the  whole  ring  or  any  part  of  it  being  pre- 
cipitated on  the  body  of  the  planet.  Every  particle  of  the  ring  is  now  to  be 
regarded  as  a  satellite  of  Saturn,  disturbed  by  the  attraction  of  a  ring  of 
satellites  at  the  same  mean  distance  from  the  planet,  each  of  which  however  is 
subject  to  slight  displacements.  The  mutual  action  of  the  parts  of  the  ring  will 
be  so  small  compared  with  the  attraction  of  the  planet,  that  no  part  of  the 
ring  can  ever  cease  to  move  round  Saturn  as  a  satellite. 

But  the  question  now  before  us  is  altogether  different  from  that  relating  to 
the  solid  ring.  We  have  now  to  take  account  of  variations  in  the  form  and 
arrangement  of  the  parts  of  the  ring,  as  well  as  its  motion  as  a  whole,  and 
we  have  as  yet  no  security  that  these  variations  may  not  accumulate  till  the 
ring  entirely  loses  its  original  form,  and  collapses  into  one  or  more  satellites, 
circulating  round  Saturn.  In  fact  such  a  result  is  one  of  the  leading  doctrines 
of  the  "  nebular  theory "  of  the  formation  of  planetary  systems :  and  we  are 
familiar  with  the  actual  breaking  up  of  fluid  rings  under  the  action  of  "capil- 
lary "  force,  in  the  beautiful  experiments  of  M.  Plateau. 

In  this  essay  I  have  shewn  that  such  a  destructive  tendency  actually  exists, 
but  that  by  the  revolution  of  the  ring  it  is  converted  into  the  condition  of 
dynamical  stability.  As  the  scientific  interest  of  Saturn's  Rings  depends  at 
present  mainly  on  this  question  of  their  stability,  I  have  considered  their  motion 
rather  as  an  illustration  of  general  principles,  than  as  a  subject  for  elaborate 
calculation,  and  therefore  I  have  confined  myself  to  those  parts  of  the  subject 
which  bear  upon  the  question  of  the  permanence  of  a  given  form  of  motion. 

There  is  a  very  general  and  very  important  problem  in  Dynamics,  the  solu- 
tion of  which  would  contain  all  the  results  of  this  Essay  and  a  great  deal 
more.     It  is  this — 

"Having  found  a  particular  solution  of  the  equations  of  motion  of  any 
material    system,  to   determine   whether   a   slight  disturbance  of  the  motion  indi- 


296  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

cated    by    the    solution    would    cause    a    small    periodic    variation,    or    a    total 
derangement   of  the   motion." 

The  question  may  be  made  to  depend  upon  the  conditions  of  a  maximum 
or  a  minimum  of  a  function  of  many  variables,  but  the  theory  of  the  tests 
for  distinguishing  maxima  from  minima  by  the  Calculus  of  Variations  becomes 
so  intricate  when  applied  to  functions  of  several  variables,  that  I  think  it  doubt- 
ful whether  the  physical  or  the  abstract  problem  will  be  first  solved. 


PART   I. 

ON   THE   MOTION   OF   A   RIGID   BODY   OF   ANY    FORM   ABOUT   A   SPHERE. 

We  confine  our  attention  for  the  present  to  the  motion  in  the  plane  of 
reference,  as  the  interest  of  our  problem  belongs  to  the  character  of  this  motion, 
and  not  to  the  librations,  if  any,  from  this  plane. 

Let  S  (Fig.  2)  be  the  centre  of  gravity  of  the  sphere,  which  we  may  call 
Satiu-n,  and  E  that  of  the  rigid  body,  which  we  may  call  the  Ring.  Join  RS, 
and  divide  it  in  G  so  that 

SG  :  GR  '.:  R  :  S, 

R  and  S  being  the  masses  of  the  Ring  and  Saturn  respectively. 

Then  G  will  be  the  centre  of  gravity  of  the  system,  and  its  position  will 
be  unaffected  by  any  mutual  action  between  the  parts  of  the  system.  Assume  G 
as  the  point  to  which  the  motions  of  the  system  are  to  be  referred.  Draw  GA 
in  a  direction  fixed  in  space. 

Let  AGR  =  e,  and  SR  =  r, 

then  ^^^'S+R^'   ^^^    ^^^STR^' 

so  that  the  positions  of  S  and  R  are  now  determined. 

Let  BRR  be  a  straight  line  through  R,  fixed  with  respect  to  the  substance 
of  the  ring,  and  let  BRK=^. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS.  297 

This  determines  the  angular  position  of  the  ring,  so  that  from  the  values 
of  r,  6,  and  ^  the  configuration  of  the  system  may  be  deduced,  as  far  as  relates 
to  the  plane  of  reference. 

We  have  next  to  determine  the  forces  which  act  between  the  ring  and 
the  sphere,  and  this  we  shall  do  by  means  of  the  potential  function  due  to 
the  ring,  which  we  shall  call   V. 

The  value  of  V  for  any  point  of  space  S,  depends  on  its  position  relatively 
to  the  ring,  and  it  is  found  from  the  equation 


where  dm  is  an  element  of  the  mass  of  the  ring,  and  r  is  the  distance  of  that 
element  from  the  given  point,  and  the  summation  is  extended  over  every  element 
of  mass  belonging  to  the  ring.  V  will  then  depend  entirely  upon  the  position 
of  the  point  S  relatively  to  the  ring,  and  may  be  expressed  as  a  function 
of  r,  the  distance  of  S  from  R,  the  centre  of  gravity  of  the  ring,  and  ^,  the 
angle  which  the  line  SR  makes  with  the  line  RB,  fixed  in  the  ring. 

A  particle   P,   placed   at   S,   will,   by   the   theory  of  potentials,    experience  a 

dV    .  ...  .  \  dV 

moving    force    P  —p    in   the   direction    which   tends   to   increase   r,    and  P  -  -jj 

in  a  tangential  direction,  tending  to  increase  ^. 

Now    we    know    that    the    attraction  of  a   sphere   is   the   same  as   that   of 

a  particle   of  equal  mass   placed    at   its   centre.      The   forces   acting  between  the 

dV  .  . 

sphere  and  the  ring  are  therefore  S  -j~  tending  to  increase  r,  and  a  tangential 

\  dV  . 

force  S  -  -j-r ,   applied   at  S  tending   to  increase  <;^.     In   estimating  the  efiect  of 

this   latter   force  on   the  ring,  we  must  resolve  it  into  a  tangential  force  S  -  -jj- 

dV 
acting  at  R,  and  a  couple  S  -j-r  tending  to  increase  (f). 

We   are   now  able   to  form   the   equations  of  motion  for  the  planet  and  the 
ring. 


298  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

For  the  planet 


^d   jf  Rr  Ydd\          _R_  ^Jy  ,  . 

^  dt  ]S^VRl  dtj  '-  "  S  +  R  '^  d<f> ^'^' 

«l(^)-^(f)'=^^' (^)- 

For  the  centre  of  gravity  of  the  ring, 

j.d   (f   Sr  Y  ^^1  S        dV  ,  . 

^dt\\S+-R)  Ttr~STR^df  ^  ^' 

j.d^  f  Sr  \  Sr     (d0Y_     dV  ,  . 

For  the  rotation  of  the  ring  about  its  centre  of  gravity, 

^S(''+«=^f (5)' 

where  h  is  the  radius  of  gyration  of  the  ring  about  its  centre  of  gravity. 

Equation  (3)  and  (4)  are  necessarily  identical  with  (l)  and  (2),  and  shew 
that  the  orbit  of  the  centre  of  gravity  of  the  ring  must  be  similar  to  that 
of  the  Planet.  Equations  (1)  and  (3)  are  equations  of  areas,  (2)  and  (4)  are 
those  of  the  radius  vector. 

Equations  (3),  (4)  and  (5)  may  be  thus  written, 

M-'^T!-'-'^}-(^-^i'- («)' 

-{§-©}-(---)f    - (^)- 

-(f-^f)--^  - («)• 

These  are  the  necessary  and  sufficient  data  for  determining  the  motion  of 
the  ring,  the  initial  circumstances  being  given. 

Prob.  I.  To  find  the  conditions  under  which  a  uniform  motion  of  the 
ring  is  possible. 

By  a  uniform  motion  is  here  meant  a  motion  of  uniform  rotation,  during 
which  the  position  of  the  centre  of  the  Planet  with  respect  to  the  ring  does 
not  change. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  299 

In    this    case    r    and    </>    are   constant,    and  therefore    V  and  its   differential 
coefficients  are  given.     Equation  (7)  becomes, 

which  shews  that  the  angular  velocity  is  constant,  and  that 

dey        R+S  dV     ,  ,^. 

r-  =  <o\  say  (9). 


dtj  Rr      dr 

(PB 
Hence,  -71  =  0,  and  therefore  by  equation  (8), 

%-^ • •■•••(-)• 

Equations  (9)  and  (10)  are  the  conditions  under  which  the  uniform  motion 
is  possible,  and  if  they  were  exactly  fulfilled,  the  uniform  motion  would  go  on 
for  ever  if  not  disturbed.  But  it  does  not  follow  that  if  these  conditions  were 
nearly  fulfilled,  or  that  if  when  accurately  adjusted,  the  motion  were  slightly 
disturbed,  the  motion  would  go  on  for  ever  nearly  uniform.  The  effect  of  the 
disturbance  might  be  either  to  produce  a  periodic  variation  in  the  elements 
of  the  motion,  the  ampUtude  of  the  variation  being  small,  or  to  produce  a 
displacement  which  would  increase  indefinitely,  and  derange  the  system  altogether. 
In  the  one  case  the  motion  would  be  dynamically  stable,  and  in  the  other  it 
would  be  dynamically  unstable.  The  investigation  of  these  displacements  while 
still  very  small  wiU  form  the  next  subject  of  inquiry. 

Prob.  II.     To  find  the  equations  of  the  motion  when  slightly  disturbed. 
Let    r  =  r„  0  =  o}t  and  (f)  =  (f>^  in.  the  case  of  uniform  motion,  and  let 

r=ro  +r„ 

e=a)t+e„ 

when   the    motion   is   slightly   disturbed,    where   r^,  6^,  and   ^1   are  to   be   treated 

as   small   quantities   of  the   first  order,  and  their   powers  and  products  are  to  be 

dV  dV 

neglected.     We  may  expand  -j-^  and  -j-r  by  Taylor's  Theorem, 

dV_dV      drV  d'V 

dr  ~dr  "^  di^    '''"*■  cZrc/t^"^^' 

d<f>~'d<f'^drd<t>''''^  d<i>''^'' 


300  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

where  the   values   of   the   differential  coeflBcients   on   the  right-hand   side   of    the 
equations  are  those  in  which  i\  stands  for  r,  and  ^^  for  ^. 

CaJlmg  ^=A    ^^^  =  M^   ^^=N, 

and  taking  account  of  equations  (9)  and  (10),  we  may  write  these  equations, 

a^= -sirs'" +^''+^^" 

Substituting  these  values  in  equations  (6),  (7),  (8),  and  retaining  all  small 
quantities  of  the  first  order  while  omitting  their  powers  and  products,  we  have 
the  following  system  of  linear  equations  in  r^,  O^,  and  ^i, 

E  (2r,co^  +  r,^^^y{E  +  S)(Mr,  +  N<f.,)    =0 (11), 


R 


d%     ,     „     de\ 


df 


(o%-2r,(o-^]-{R  +  S){L7\  +  M<f>,)  =  0 (12), 


RlH'^^  +  ^-SiMr^  +  N^:)  =0 (13). 


df   '  df 


Prob.    III.     To   reduce   the   three   simultaneous   equations   of   motion  to  the 
form  of  a  single  linear  equati 


:ion. 


Let  us  write  n  instead  of  the  symbol  -j- ,  then  arranging  the  equations  in 
terms  of  i\,  6^,  and  j>^,  they  may  be  written: 

{2R,o>n  +  (R  +  S)M}r,  +  (Rr:n')e,  +  {R  +  S)N<i>,         =0 (14), 

{Rn'-R<^'^-(R  +  S)  L}r,-(2Rr,con)d,^{R  +  S)M<f>,  =  0 (15), 

-  (SM)  r,  +  (Rk'n')  0,  +  {RUrv  -SN)<j>,        =0 (16). 

Here  we  have  three  equations  to  determine  three  quantities  r,,  6„  ^i ;  but 
it  is  evident  that  only  a  relation  can  be  determined  between  them,  and  that 
in  the  process  for  finding  their  absolute  values,  the  three  quantities  will  vanish 
together,  and  leave  the  following  relation  among  the  coefiicients, 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN'S    RINGS.  301 

-{2Rr,oin+  {R  +  S)^r}  [2R)\(on]  [Rlcrc'-SN} 
+  {Rn'  -  Rco'  -(R  +  S)  L]  {Rh'rf}  {(R  +  >S')  N] 

+  {SM)  {Rrjn')  {R  +  S)M-  (SM)  {2Rr,<on)  (R  +  S)Xi=0 (17). 

+  {2Rr,<on  +  (R  +  S)M}  {RLni'}  {(R  +  S)  if} 
-  {Rn'  -  Rxo'  -{R  +  S)}  {Rr.'if}  {RJc'n'  -  SN} 
By    multiplying    up,    and   arranging    by   powers   of    n   and   dividing   by   Rn\ 
this  equation  becomes 

Aii*  +  B)v+C=0 (18), 

where 

B  =  SRr-r:i''<o-'-R{R  +  S)Lr:Jc'-R{{R  +  S)]if  +  Si''}N-  i (19). 

C=R{(R  +  S)l''-  3Sr:}  oy  +  (R  +  S)  {{R  +  S)  t  +  Sr^}  (Z.V-  IP)  J 
Here    we    have    a    biquadratic    equation   in   ?i   which   may   be    treated   as  a 
quadratic  in  ?r,  it  being  remembered  that  ?i  stands  for  the  operation  -j-  . 

Prob.    IV.      To    determine    whether    the    motion    of    the   ring  is   stable    or 
unstable,  by  means  of  the  relations  of  the  coefficients  A,  B,  C. 

The  equations  to  determine  the  forms  of  r^,  6^,  and  <^i  are  all  of  the  form 
.  d*u     -r,  dhi     ^       ^  /^^\ 

^*+-^*+^"=» (-°'' 

and  if  n  be  one  of  the  four  roots  of  equation  (18),  then 

will   be   one   of    the   four   terms   of   the   solution,    and   the   values   of  i\,  6^,   and 
<^i  will  differ  only  in  the  values  of  the  coefficient  D. 

Let   us  inquire  into  the  nature  of  the  solution  in  different  cases. 

(1)  If    n    be    positive,    this    term   would   indicate   a   displacement   which 
must  increase  indefinitely,  so  as  to  destroy  the  arrangement  of  the  system. 

(2)  If  n  be  negative,   the  disturbance  which  it  belongs  to  would  gradually 
die  away. 

(3)  If  n   be  a  pure  impossible  quantity,  of  the  form    ±aj  —\,    then  there 
will   be   a   term   in   the   solution   of  the  form  D  cos  [at  +  a),  and  this   would  indi- 

277 

cate  a  periodic  variation,  whose  amplitude  is  D,  and  period  ^^ . 


302  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

(4)  If    n   be   of   the   form   b±J'^a,   the   first   term   being   positive   and 
the  second  impossible,  there  will  be  a  term  in  the  solution  of  the  form 

De^'  cos  {at  +  a), 
which    indicates    a    periodic    disturbance,    whose    amplitude   continually  increases 
till  it  disarranges  the  system. 

(5)  If  n  be  of   the   form    -h±s/-la,   a   negative  quantity   and   an   im- 
possible one,  the  corresponding  term  of  the  solution  is 

i>e"*'cos  {(it  +  a), 
which  indicates  a  periodic  disturbance  whose  amplitude  is  constantly  diminishing. 

It  is  manifest  that  the  first  and  fourth  cases  are  inconsistent  with  the 
permanent  motion  of  the  system.  Now  since  equation  (18)  contains  only  even 
powers  of  n,  it  must  have  pairs  of  equal  and  opposite  roots,  so  that  every 
root  coming  under  the  second  or  fifth  cases,  implies  the  existence  of  another 
root  belonging  to  the  first  or  fourth.  If  such  a  root  exists,  some  disturbance 
may  occur  to  produce  the  kind  of  derangement  corresponding  to  it,  so  that 
the  system  is  not  safe  unless  roots  of  the  first  and  fourth  kinds  are  altogether 
excluded.  This  cannot  be  done  without  excluding  those  of  the  second  and  fifth 
kinds,  so  that,  to  insure  stability,  aU  the  four  roots  must  be  of  the  third  kind, 
that  is,  pure  impossible  quantities. 

That  this  may  be  the  case,  both  values  of  n"  must  be  real  and  negative, 
and  the  conditions  of  this  are — 

1st.     That  A,  B,  and  C  should  be  of  the  same  sign, 

2ndly.     That  R>iAC. 

When  these  conditions  are  fulfilled,  the  disturbances  will  be  periodic  and 
consistent  with  stability.  When  they  are  not  both  fulfilled,  a  small  disturbance 
may  produce  total  derangement  of  the  system. 

Prob.  V.  To  find  the  centre  of  gravity,  the  radius  of  gyration,  and  the 
variations  of  the  potential  near  the  centre  of  a  circular  ring  of  small  but  variable 
section. 

Let  a  be  the  radius  of  the  ring,  and  let  6  be  the  angle  subtended  at  the 
centre  between  the  radius  through  the  centre  of  gravity  and  the  line  through 
a  given  point   in   the  ring.     Then   if  /i   be   the   mass    of  unit    of  length   of  the 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURn's    RINGS.  303 

ring   near    the  given    point,   ft   will   be  a  periodic   function   of  6,    and  may    there- 
fore be  expanded  by  Fourier's  theorem  in  the  series, 

li  =  —  {1  + 2/cos^  +  §^cos2^  +  §/isin2^  +  2ico3(3^  +  a)  +  &c.} (21), 

where/,  g,  h,  &c.  are  arbitrary  coefficients,  and  R  is  the  mass  of  the  ring. 

(1)  The  moment  of  the  ring  about  the  diameter  perpendicular  to  the 
prime  radius  is 

R)\=  r  ficr  cos  ecW  =  Raf, 
therefore  the  distance  of  the  centre  of  gravity  from  the  centre  of  the  ring, 

(2)  The  radius  of  gyration  of  the  ring  about  its  centre  in  its  own  plane 
is  evidently  the  radius  of  the  ring  =a,  but  if  k  be  that  about  the  centre  of 
gravity,  we  have 

.'.  Af  =  a=(l-f). 

(3)  The  potential  at  any  point  is  found  by  dividing  the  mass  of  each 
element  by  its  distance  from  the  given  point,  and  integrating  over  the  whole 
mass. 

Let   the  given  point  be  near  the  centre  of  the  ring,  and  let  its  position  be 
defined  by  the  co-ordinates  r    and  xjj,  of  which  r   is  small  compared  with  a. 

The  distance  (p)  between  this  point  and  a  point  in  the  ring  is 
i  =  i  {1  +  %03  (^  -  0)  +  i  (Q'  + 1  (3'  cos  2{i,-0)+&c.}. 


The  other  terms  contain  powers  of  —  higher  than  the  second. 
We  have  now  to  determine  the  value  of  the  integral, 

Jo    P 
and    in   multiplying    the    terms    of    (/i)    by   those   of    f-J  ,   we   need   retain   only 
those    which    contain    constant    quantities,    for   all   those   which   contain   sines  or 


304  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

cosines   of  multiples   of  {^1^  —  0)   will   vanisti   when  integrated  between  the   limits. 
In  this  way  we  find 

^=-  {l+/%osr/;  +  i^'(l-4-5rcos2i/,  +  ^sin2tA)} (22). 

The  other  terms  containing  higher  powers  of  — . 

In   order  to  express   V  in  terms  of  r,  and  (f)„   as   we  have  assumed   in   the 
former  investigation,  we  must  put 

r'  C09  xjj=  —  Tj  +  ^r^^/, 

^=§{^-f'i^it^^+9)  +  i^fr.<f>.  +  ir<l>n^-9)} (23). 


From  which  we  find  ,    , 
dr 


'^.-s^- 


S.='^=i'(i+^) 


K).=^=i^'(^-^) 


These  results  may  be  confirmed  by  the  following  considerations  applicable  to 
any  circular  ring,  and  not  involving  any  expansion  or  integration.  Let  af  be 
the  distance  of  the  centre  of  gravity  from  the  centre  of  the  ring,  and  let 
the  ring  revolve  about  its  centre  with  velocity  o).  Then  the  force  necessary 
to  keep  the  ring  in  that  orbit  will  be   —Rafoi^. 

But  let  >S  be  a  mass  fixed  at  the  centre  of  the  ring,  then  if 

a' 
every  portion   of  the  ring  will  be  separately   retained   in  its  orbit  by   the  attrac- 
tion of  S,   so  that  the   whole   ring   will  be  retained  in  its   orbit.     The   resultant 
attraction  must  therefore  pass  through  the  centre  of  gravity,  and  be 

-^  a} 

therefore  ^^^rL, 

dr  a: 


ON    THE    STABILITY     OF    THE    MOTION    OF    SATURN's    RINGS.  305 

cl'V     cfV     d'V 
The  equation  3^  +  rf^-  +  dz'  +  *'P  =  « 

is  true  for  any  system  of  matter  attracting  according  to  the  law  of  gravitation. 
If  we  bear  in  mind  that  the  expression  is  identical  in  form  with  that  which 
measures  the  total   efflux   of   fluid   from   a  differential  element  of  volume,  where 

-J-  ,   -J- ,  -7-   are   the  rates  at  which  the  fluid  passes  through  its  sides,  we  may 

easily  form  the  equation  for  any  other  case.  Now  let  the  position  of  a  point 
in  space  be  determined  by  the  co-ordinates  r,  ^  and  z,  where  z  is  measured 
perpendicularly  to  the  plane  of  the  angle  <j>.  Then  by  choosing  the  directions 
of  the  axes  x,  y,  z,  so  as  to  coincide  with  those  of  the  radius  vector  r,  the  per- 
pendicular to  it  in  the  plane  of  <^,  and  the  normal,  we  shall  have 
dx  =  dr^        dy  =  rd^,  dz  =  dz, 

dV^dV     dV^ldV     dV^dV 
dx~  dr  ^    dy      r  d<l>'    dz       dz 

The  quantities  of  fluid  passing  through  an  element  of  area  in  each  direction  are 
-T-  rd(paz,     -j-7  -  ardz,     -p  rdcpdr, 

so  that  the  expression  for  the  whole  efflux  is 

1  dV    d^V     1    d^V    d^V 

r   dF^d^^7    df^d^ ^^^' 

which  is  necessarily  equivalent  to  the  former  expression. 

d^V 
Now  at  the  centre  of  the  ring  -r^  may  be  found  by  considering  the  attrac- 
tion on  a  point  just  above  the  centre  at  a  distance  z, 

dV_      p       z 

dz  {a'->tz'f' 

d'V        R       . 

-^=--3,whenz  =  0. 

Ai  1  \  dV        R        ,  . 

Also  we  know  ^  =  — ^ ,  and  r  =  aj, 

V     (XV  (Xi 

so  that  m  any  curcular  rmg  "^^^^  d^^    a^ ^     ** 

an  equation  satisfied  by  the  former  values  of  L  and  N. 


306  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

By  referring  to  tlae  original  expression  for  the  variable  section  of  the  ring, 
it  appears  that  the  effect  of  the  coefficient  /  is  to  make  the  ring  thicker  on 
one  side  and  thinner  on  the  other  in  a  uniformly  graduated  manner.  The  eflfect 
of  ^  is  to  thicken  the  ring  at  two  opposite  sides,  and  diminish  its  section  in 
the  parts  between.  The  coefficient  h  indicates  an  inequality  of  the  same  kind, 
only  not  symmetrically  disposed  about  the  diameter  through  the  centre  of 
gravity. 

Other  terms  indicating  inequalities  recurring  three  or  more  times  in  the 
circumference  of  the  ring,  have  no  effect  on  the  values  of  X,  M  and  N.  There  is 
one  remarkable  case,  however,  in  which  the  irregularity  consists  of  a  single 
heavy  particle  placed  at  a  point  on  the  circumference  of  the  ring. 

Let  P  be  the  mass  of  the  particle,  and  Q  that  of  the  uniform  ring  on 
which  it  is  fixed,  then  R  =  P-{-Q, 

■>     K' 


-^S-^.=^(-^S=.4(-^) 


•••  3  =  ^  =  3/- (27)- 

Prob.  VI.  To  determine  the  conditions  of  stability  of  the  motion  in  terms 
of  the  coefficients/,  g,  h,  which  indicate  the  distribution  of  mass  in  the  ring. 

The  quantities  which  enter  into  the  differential  equation  of  motion  (18) 
are  R,  S,  k",  i\,  (o",  L,  M,  N.  We  must  observe  that  S  is  very  large  compared 
with  R,  and  therefore  we  neglect  R  in  those  terms  in  which  it  is  added  to  S, 
and  we  put 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS.  307 

Substituting  these  values  in  equation  (18)  and  dividing  by  H'a*/-,  we  obtain 
{l-P)n*  +  (l-y^  +  y^g)nW  +  (^-&r-lg^-lh^  +  2fg)<.^  =  0 (28). 

The  condition  of  stability  is  that  this  equation  shall  give  both  values  of  n* 
negative,  and  this  renders  it  necessary  that  all  the  coefficients  should  have  the 
same  sign,  and  that  the  square  of  the  second  should  exceed  four  times  the 
product  of  the  first  and  third. 

(1)  Now  if  we  suppose  the  ring  to  be  uniform,  /,  g  and  h  disappear, 
and  the  equation  becomes 

n'  +  nV  +  |  =  0 (29), 

which  gives  impossible  values  to  n'  and  indicates  the  instability  of  a  uniform 
ring. 

(2)  If  we  make  g  and  A  =  0,  we  have  the  case  of  a  ring  thicker  at  one 
side  than  the  other,  and  varying  in  section  according  to  the  simple  law  of  sines. 
We  must  remember,  however,  that  /  must  be  less  than  ^,  in  order  that  the 
section  of  the  ring  at  the  thinnest  part  may  be  real.     The  equation  becomes 

(l_/=),,*  +  (l.|/^)^V  +  (|-6/>*  =  0 (30). 

The  condition  that  the  third  term  should  be  positive  gives 

/*<'375. 

The  condition  that  n'  should  be  real  gives 

71/^-112/^  +  32  negative, 

which  requires/"  to  be  between  "37445  and  1'2. 

The  condition  of  stability  is  therefore  that  /^  should  lie  between 

•37445  and  '375, 

but  the  construction  of  the  ring  on  this  principle  requires  that  /-  should  be 
less  than  "25,  so  that  it  is  impossible  to  reconcile  this  fonn  of  the  ring  with 
the  conditions  of  stability. 

(3)  Let  us  next  take  the  case  of  a  uniform  ring,  loaded  with  a  heavy 
particle  at  a  point  of  its  circumference.  We  have  then  g  =  Sf,  h  =  0,  and  the 
equation  becomes 

(l-/=)n^  +  (l-|/^  +  f/ViV+(|-y/'+6/>^  =  0 (31). 


308  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  3    RINGS. 

Dividing  each  term  by  1  -/,  we  get 

(l+/)n^+(l+/-f/0^^V  +  f{3(l+/)-8/=}a,^  =  O (32). 

The  first  condition  gives /less  than  '8279. 

The  second  condition  gives  /  greater  than  '8 15865. 

Let  us  assume  as  a  particular  case  between  these  limits  /=  •82,  which 
makes  the  ratio  of  the  mass  of  the  particle  to  that  of  the  ring  as  82  to  18, 
then  the  equation  becomes 

l-82  7i^  +  '8114?iV+-9696a>'  =  0  (33), 

which  gives  >J^^n=  ±'5916(o  or  ±-3076w. 

These  values  of  n  indicate  variations  of  r^,  O^,  and  ^i,  which  are  com- 
pounded of  two  simple  periodic  inequalities,  the  period  of  the  one  being  1"69 
revolutions,  and  that  of  the  other  3 '2  51  revolutions  of  the  ring.  The  relations 
between  the  phases  and  ampUtudes  of  these  inequalities  must  be  deduced  from 
equations  (14),  (15),  (16),  in  order  that  the  character  of  the  motion  may  be 
completely  determined. 

Equations  (14),  (15),  (16)  may  be  written  as  follows: 

{Anco  +  hoi')  ^  +2f7i%+f(3-g) (o"'(l>,  =  0 (34), 

{ii^-l<o'^{S+g)}^'-2fcone,^ifh<o'<f>,  =  0 (35), 

-/ho>^  '^  +  2  (1  -f^)n%  +  {2  (1  -f)  n'-r  {S-g)  co^}<l>,  =  0 (36). 

By  eliminating  one  of  the  variables  between  any  two  of  these  equations, 
we  may  determine  the  relation  between  the  two  remaining  variables.  Assuming 
one  of  these  to  be  a  periodic  function  of  t  of  the  form  A  cos  pt,  and  remem- 
bering that  n  stands  for  the  operation  -7- ,  we  may  find  the  form  of  the  other. 

Tlius,  eliminating  6^  between  the  first  and  second  equations, 

{n'  +  i7i<o'{5-g)  +  hoj'f-^+foy'{{3-g)<o-ym}cf>,  =  0 (37). 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  309 

T 

Assuming  —  =A^\wvt^  and  <f)i  =  Q  cos  (ut  —  ^), 

{-v'  +  ^vo)'  (5  -  g)}  A  cos  pt  +  h(o^  A  sin  vt  +fo/  (3  -rj)  Qcos{vt  -  /3)  +  Ifhui'vQ  sin  {yt  -  /3). 
Equating  vt  to  0,  and  to  -  ,  we  get  the  equations 

[v'-^voy  (5  -g)}  A  =f<o'Q  {(3  -g)  cj  cos  /8 - ^/ii/ sin /3}, 

-  h<o'  A  =fo)'Q  {(3  -  </)  o)  sin  /8  +  -l/ii/  cos  ^8}, 

from  which  to  determine  Q  and  ^. 

In  all  cases  in  which  the  mass  is  disposed  symmetrically  about  the  diameter 
through  the  centre  of  gravity,  A  =  0  and  the  equations  may  be  greatly  simplified. 

Let  6i  =  P  cos  (vt  — a),  then  the  second  equation  becomes 

{v'  +  ^0)'  (3  +  g)}  A  sin  vt  =  2Pfa}v  sin  {vt  -  a), 

whence  a  =  0,   P  =  ^^JtMiijO  .4  (38). 

2j(DV  ^        ' 

The  first  equation  becomes 

^Aoiv  cos  vt  -  2Pfv-  cos  vt  +  Qf  (3  -g)  w'  cos  (I'f  -  /S)  =  0, 

whence  ^  =  0,   <?  =  '^"t.f '  w^^-^  (S^)- 

In  the   numerical  example  in  which  a  heavy  particle  was  fixed  to   the   cir- 
cumference of  the  ring,  we  have,  when  /=  '82, 


V 

^      1-3076 


/•5916         P_r3-21         Q_f-l-229 
t-3076'      A~\b-72'      A~\-   797' 

so  that  if  we  put  (ot  =  0^  =  the  mean  anomaly, 

^  =  .4sin(-5916(9o-a)+^sin(-3076  6'o-^)  (40), 


^1  =  3-21^  cos  (-5916(90- a) +  5-72^  cos  (-3070  ^0-/3) (41), 

<^,=  -l-229^cos(-5916l9o-a)-5-7975cos(-30766',-/3)  ...  (42). 

These   three   equations    serve    to    determine  1\,   6^  and  <^i  when   the  original 
motion  is   given.      They  contain  four  arbitrary  constants  A,  B,  a,  /3.     Now  since 


310  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

the  original  values  1\,  0^,  <^i,  and  also  their  first  differential  coefficients  with 
respect  to  t,  are  arbitrary,  it  would  appear  that  six  arbitrary  constants  ought 
to  enter  into  the  equation.  The  reason  why  they  do  not  is  that  we  assume 
r„  and  0^  as  the  Tiiean  values  of  r  and  6  in  the  actucd  motion.  These  quantities 
therefore  depend  on  the  original  circumstances,  and  the  two  additional  arbitrary 
constants  enter  into  the  values  of  ^o  and  d^.  In  the  analytical  treatment  of  the 
problem  the  differential  equation  in  n  was  originally  of  the  sixth  degree  with  a 
solution  n-  =  0,  which  implies  the  possibihty  of  terms  in  the  solution  of  the 
form  Ct  +  D. 

The  existence  of  such  terms  depends  on  the  previous  equations,  and  we  find 
that  a  term  of  this  form  may  enter  into  the  value  of  6,  and  that  r^  may  contain 
a  constant  term,  but  that  in  both  cases  these  additions  will  be  absorbed  into 
the  values  of  0,  and  r,. 


PART    IL 

ON    THE    MOTION   OF   A   RING,    THE   PARTS   OF   WHICH   ARE  NOT   RIGIDLY   CONNECTTED. 

1.  In  the  case  of  the  Ring  of  invariable  form,  we  took  advantage  of  the 
principle  that  the  mutual  actions  of  the  parts  of  any  system  form  at  all  times 
a  system  of  forces  in  equilibrium,  and  we  took  no  account  of  the  attraction 
between  one  part  of  the  ring  and  any  other  part,  since  no  motion  could  result 
from  this  kind  of  action.  But  when  we  regard  the  different  parts  of  the  ring 
as  capable  of  independent  motion,  we  must  take  account  of  the  attraction  on 
each  portion  of  the  ring  as  affected  by  the  irregularities  of  the  other  parts,  and 
therefore  we  must  begin  by  investigating  the  statical  part  of  the  problem  in 
order  to  determine  the  forces  that  act  on  any  portion  of  the  ring,  as  depending 
on  the  instantaneous  condition  of  the  rest  of  the  ring. 

In  order  to  bring  the  problem  within  the  reach  of  our  mathematical  methods, 
we  limit  it  to  the  case  in  which  the  ring  is  nearly  circular  and  uniform,  and  has 
a  transverse  section  very  small  compared  with  the  radius  of  the  ring.  By 
analysing  the  difficulties  of  the  theory  of  a  linear  ring,  we  shall  be  better  able 
to  appreciate  those  which  occur  in  the  theory  of  the  actual  rings. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  311 

The  ring  which  we  consider  is  therefore  small  in  section,  and  very  nearly 
circular  and  uniform,  and  revolving  with  nearly  uniform  velocity.  The  variations 
from  circular  form,  uniform  section,  and  uniform  velocity  must  be  expressed  by  a 
proper  notation. 

2.  To  express  the  position  of  an  element  of  a  variable  ring  at  a  given  time 
in  terms  of  the  original  position  of  the  element  in  the  ring. 

Let  S  (fig.  3)  be  the  central  body,  and  SA  a  direction  fixed  in  space. 

Let  SB  be  a  radius,  revolving  with  the  mean  angular  velocity  w  of  the 
ring,  so  that  ASB  =  (ot. 

Let  n  be  an  element  of  the  ring  in  its  actual  position,  and  let  P  be  the 
position  it  would  have  had  if  it  had  moved  uniformly  with  the  mean  velocity  w 
and  had  not  been  displaced,  then  BSP  is  a  constant  angle  =s,  and  the  value 
of  5  enables  us  to  identify  any  element  of  the  ring. 

The  element  may  be  removed  from  its  mean  position  P  in  three  different 
ways. 

(1)  By  change  of  distance  from  S  by  a  quantity  l^TT  =  p. 

(2)  By  change  of  angular  position  through  a  space  Pp  =  a. 

(3)  By  displacement  perpendicular  to  the  plane  of  the  paper  by  a  quantity  C 

p,  a-  and  ^  are  all  functions  of  s  and  t.  If  we  could  calculate  the  attrac- 
tions on  any  element  as  depending  on  the  form  of  these  functions,  we  miglit 
determine  the  motion  of  the  ring  for  any  given  original  disturbance.  We  cannot, 
however,  make  any  calculations  of  this  kind  without  knowing  the  form  of  the 
functions,  and  therefore  we  must  adopt  the  following  method  of  separating  the 
original  disturbance  into  others  of  simpler  form,  first  given  in  Fourier's  Tmitc 
de  Chaleur. 

3.  Let  C/"  be  a  function  of  s,  it  is  required  to  express  U  in  a  series  of 
sines  and  cosines  of  multiples  of  s  between  the  values  5  =  0  and  .s  =  2t. 

Assume  U=A,coss  +  A.,  cos  2*-  +  &c.  -f  A ^  cos  nis  +  A „  cos  ns 

+  B,  sin  ,s  +  B,  cos  2.s  +  &c.  +  B,„  sin  ms  +  B„  sin  ns. 


312  ON    THE    STABILITY    OF    THE    MOTION    OP    SATURN's    RINGS. 

Multiply  by  coa  Tusds  and  integrate,  then  all  terms  of  the  form 
J  cos  ms  cos  nsds  and  /  cos  ms  sin  nsds 
will  vanish,  if  we  integrate  from  s  =  0  to  s  =  27r,  and  there  remains 

I    U  COS  msds= IT  A^,  Ua\-D.msds  =  'TrB^. 

If  we  can  determine  the  values  of  these  integrals  in  the  given  case,  we 
can  find  the  proper  coefficients  A^,  B^,  &c.,  and  the  series  will  then  represent 
the  values  of  U  from  s  =  0  to  5  =  27r,  whether  those  values  be  continuous  or 
discontinuous,  and  when  none  of  those  values  are  infinite  the  series  will  be 
convergent. 

In  this  way  we  may  separate  the  most  complex  disturbances  of  a  ring  into 
parts  whose  form  is  that  of  a  circular  function  of  s  or  its  multiples.  Each  of 
these  partial  disturbances  may  be  investigated  separately,  and  its  efiect  on  the 
attractions  of  the  ring  ascertained  either  accurately  or  approximately. 

4.  To  find  the  magnitude  and  direction  of  the  attraction  between  two 
elements  of  a  disturbed  ring. 

Let  P  and  Q  (fig.  4)  be  the  two  elements,  and  let  their  original  positions 
be  denoted  by  s^  and  5j,  the  values  of  the  arcs  BP,  BQ  before  displacement. 
The  displacement  consists  in  the  angle  BSP  being  increased  by  ctj  and  BSQ 
by  0*2 ,  while  the  distance  of  P  from  the  centre  is  increased  by  p,  and  that  of 
Q  by  Pj.  We  have  to  determine  the  effect  of  these  displacements  on  the  distance 
PQ  and  the  angle  SPQ. 

Let  the  radius  of  the  ring  be  unity,  and  5j  — .9i  =  2^,  then  the  original 
value  of  PQ  will  be  2  sin  0,  and  the  increase  due  to  displacement 

=  (/>2  +  Pi)  sin  ^  +  (o-j  -  (Ti)  cos  6. 

We  may  write  the  complete  value  of  PQ  thus, 

PQ  =  2Bme{l+i{p,  +  p,)+^{(T,-(T,)cot0\ (1). 

The  original  value  of  the  angle  SPQ  was  -^-6,   and    the    increase    due   to 

displacement  is  i{Pi  —  Pi)  cot  ^  -  ^  (o-j  -  Ci), 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  8    RINGS. 


313 


30  that  we  may  write  the  values  of  sin  SPQ  and  cos  SPQ, 

Gin  SFQ  =  cos  e {I +i{p,-p,)-i  {a-,- a,)  ta,n0}  (2), 

cos  SPQ  =  am  e  {I -i(p,-p,)coVd  +  i  (a-,- a-,)  cot  6}  (3). 


If  we   assume   the    masses  of  P  and  Q  each  equal  to  -  R,  where  P   is   the 

mass   of  the   ring,  and  p,  the   number  of  satellites   of  which   it   is   composed,  the 
accelerating  effect  of  the  radial  force  on  P  is 

li}22^  =  l--«_^{l_(p.  +  p,)_i(p._p,)eof^-iK-.T.)cot3}...(4), 

and  the  tangential  force 

I    j^sinSPQ        li^COS^-.  ,  \  /      +  ^   ,    l  x         mi  /r:\ 

]1^     PQ     ^^H^i^I^-^/^^-f/^^-l^'-^Olcot^  +  itan^)} (5). 

1       L  —  l 
The  normal  force  is  -R  ^   .  ,\. 
p.     8  sm^  6 

5.  Let  us  substitute  for  p,  or  and  {  their  values  expressed  in  a  series  of 
sines  and  cosines  of  multiples  of  5,  the  terms  involving  ms  being 

Pi  =  A  cos  {ms  +  a),  pi  =  A  cos  (ms  +  a  +  20), 

o-,  =  -Bsin(m5  +  ^),  cr.  =  B  sin  {7}is-\-fi  + 20), 

C,  =  C  cos  (ms  +  y),  C2  =  Ccos  {ms  +  y  +  26). 

The  radial  force  now  becomes 

1  —  ^  cos  {ms  +  a)  ( 1  +  cos  2m0)  +  A  sin  {ms  +  a)  sin  2md  i 

+  ^A  cos  {ms  +  a)  (1  -  cos  2m6)  cot'  ^  -  ^^  sin  (t/i^  +  a)  sin  2ni6  cot" 6  \  (6). 
+^B  sin  {ms  + ft)  {1  -cos  2m^)  cot  ^-^5cos(??i5  +  /8)  siii2w^cot^. 

The  radial  component  of  the  attraction  of  a  corresponding  particle  on  the 
other  side  of  P  may  be  found  by  changing  the  sign  of  6.  Adding  the  two 
together,  we  have  for  the  effect  of  the  pair 

-  ^-^ — ^  {1  —  ^  COS  {ms  +  a)  (2  cos"  md  —  sin' md  cot'  6) 

-  B  cos  {ms  + 13)  ^  sin  2m6  cot  6] 


I_i2_ 
/x  4  sin  ^ 


(?)• 


314 


Let  us  put 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

sin*  mO  cos''  6     cos'  m6\ 


K=t 


f^va.  2m6  cos 
\      4sin*^ 

/sin"  md  cos'  6 

sin'  mt 
2  sin^ 

1 


0 


+  i 


sin'?n^ 


2sin^ 


(8)^: 


where    the    summation    extends    to    all    the    sateUites    on    the    same   side   of  F, 
that  is,   every    value   of    6   of    the   form    -  tt,    where   x   is   a   whole    number  less 


than 


The  radial  force  may  now  be  written 


P  =  ~R  {K+  LA  cos  ims  +  a)  -  MB  cos  {'tm  +  ^)} 


(9). 


*  Tlie  following  values  of  several  quantities  which  enter  into  these  investigations  are  calculated  for  a 
ring  of  36  satellites. 

A' =24-5. 


^  sin-  md  cos-  $       ^  cos^  md          ^ 

if 

.V 

sinS  d                    sin  e 

m=    0 

0         43    -  43 

0 

0 

ni=  1 

32         32    -16 

16 

37 

m=    2 

107         28     26 

25 

115 

m  —    3 

212         25     81 

28 

221 

;u=  4 

401         24    177 

32 

411 

vi=    9 

975         20    468 

30 

986 

/ft- 18 

1569         18    767 

0 

1582 

r   gi-eat, 

-  Z  -  -5259  when  m  -- 
=  '4342   „  TO  = 
=  -3287   „  m  = 

"3' 

ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  8    RINGS. 


315 


The  tangential  force  may  be  calculated  in  the  same  way,  it  is 

T=-  R{MAam(iiis-\-a)  +  NBsm(7ns  +  IB)} (10). 


The  normal  force  is 


Z=  -^-RJC  cos  (ms  +  y) (11). 


G.  We  have  found  the  expressions  for  the  forces  which  act  upon  each 
member  of  a  system  of  equal  satellites  which  originally  formed  a  uniform  ring, 
but  are  now  aflfected  with  displacements  depending  on  circular  functions.  If 
these    displacements    can    be   propagated   round   the    ring  in   the    form   of  waves 

with  the  velocity   — ,    the    quantities    a,  y8,    and   y   will   depend    on    t,    and    the 

complete   expressions  will  be 

p  =  ^  cos  (ms  +  nt-\-  a)  ' 

a  =  Bam(ms  +  nt+^)  ■ (12). 

^  =  Ccos  (ms  +  nt  +  y). 
Let    us    find    in    what    cases    expressions    such  as   these    will   be   true,    and 
what  will  be  the  result  when  they  are  not  true. 

Let  the  position  of  a  satellite  at  any  time  be  determined  by  the  values 
of  r,  (j),  and  C,  where  r  is  the  radius  vector  reduced  to  the  plane  of  reference, 
<t>  the  angle  of  position  measured  on  that  plane,  and  ^  the  distance  from  it. 
The  equations  of  motion  will  be 


[dtj       df     ^  r-^-^ 


dr  d4        d^_^ 
^Tt    dt  ^"^  df~ 

d^ 

df' 


1^ 


.(13). 


If  we  substitute  the  value  of  ^  in  the  third  equation  and  remember  that  r 
is  nearly   =  1 ,  we  find 


(14). 


As    this   expression   is   necessarily   positive,    the   value   of  n'   is   always   real, 
and    the    disturbances    normal   to    the   plane    of    the    ring   can   always  be    propa- 


31G  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

gated  as  waves,  and  therefore  can  never  be  the  cause  of  instability.  We 
therefore  confine  our  attention  to  the  motion  in  the  plane  of  the  ring  as 
deduced  from  the  two  former  equations. 

Putting  r  =  1 4-  /)  and   (f>  =  <ot  +  s  +  a;    and    omitting    powers    and    products    of 
p,  cr  and  their  differential  coeflScients, 


''+'">+2-t-t='^-2«''+^ 


-l+§=^ 


(15). 


Substituting  the  values  of  p  and  cr  as  given  above,  these  equations  become 
oi'-S--  RK+  U- -]-2S--EL  +  7f)A  cos  (ttis  +  nt  +  a) 

+  (2(071  + -RM)B  COS  (ins +  nt  +  ^)  =  0 ...(16), 

H' 

(2(071  +  - EM)  A  sin  (ins  +  nt  +  a)  +  (if +-RN)Bam(7ns  +  nt-\-^)  =  0.... (17). 
p  p 

Putting    for    (ins  +  nt)    any    two    diflferent   values,    we    find    from    the    second 
equation  (17) 

a=)8 (18), 

and  (2(on  +  -E]\f)A  +  (n'+-EN)B  =  0 (19), 

and  from  the  first  (16)  ((o' +  2S -- EL  +  iv)  A  +  (2(on  +  -  EM)  B  =  0 (20), 

and  (o'-S--EK=0 (21). 

p 

Eliminating  A  and  B  from  these  equations,  we  get 
n'-{S(o'-2S  +  -E(L-N)}n^ 

-'4(o-EMn  +  ((o'  +  2S--EL)-EN--,E'M'  =  0 (22), 

a  biquadratic  equation  to  determine  n. 

For   every  real  value   of  n  there  are  terms  in  the  expressions  for  p  and  o- 

of  the  form 

A  cos  (nis  +  nt  +  a). 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  317 

For  every  pure  impossible  root  of  the  form  ±7  —  In'  there  are  terms  of 
the  forms 

^e^^'cos  (ms  +  a). 

Although  the  negative  exponential  coefficient  indicates  a  continually  diminlshmg 
displacement  which  is  consistent  with  stability,  the  positive  value  which  neces- 
sarily accompanies  it  indicates  a  continually  increasing  disturbance,  which  would 
completely  derange  the  system  in  course  of  time. 

For  every  mixed  root  of  the  form   ±n/  — In'  +  n,  there  are  terms  of  the  form 

.46*"''  cos  {ms  +  nt  +  a). 

If  we  take  the  positive  exponential,  we  have  a  series  of  m   waves  travelling 

with   velocity   —    and    increasing    in    amplitude   with    the    coefficient    e"^"'.     The 

negative  exponential  gives  us  a  series  of  m  waves  gradually  dying  away,  but 
the  negative  exponential  cannot  exist  without  the  possibility  of  the  positive  one 
having  a  finite  coefficient,  so  that  it  is  necessary  for  the  stability  of  the  motion 
that  the  four  values  of  n  be  all  real,  and  none  of  them  either  impossible 
quantities  or  the  sums  of  possible  and  impossible  quantities. 

We  have  therefore  to  determine  the  relations  among  the  quantities  K,  L, 
M,  N,  R,  S,  that  the  equation 

n'-lS+^RidK+L-N)]?^ 

'-4<o-RMn  +  {SS+  -  R  (K-L)}  -  RN-  \  R'M'^  U=0 

may  have  four  real  roots. 

7.  In  the  first  place,  U  is  positive,  when  tz  is  a  large  enough  quantity, 
whether  positive  or  negative. 

It  is  also  positive  when  7i=;0,  provided  S  be  large,  as  it  must  be,  com- 
pared with  -  RL,  -  RM  and   -  RN. 

If  we  can  now  find  a  positive  and  a  negative  value  of  n  for  which  U 
is  negative,  there  must  be  four  real  values  of  n  for  which  U=0,  and  the  four 
roots  will  be  real. 


318  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

Now  if  we  put  n=  ±J^JS, 

U=  -^S'  +  l  -R{7N±ij2M-L-dK)  S+  \r{KN-LN^M% 

which  is  negative  if  >S  be  large  compared  to  R. 

So  that  a  ring  of  satellites  can  always  be  rendered  stable  by  increasing 
the  mass  of  the  central  body  and  the  angular  velocity  of  the  ring. 

The   values   of  L,  M,  and  N  depend  on  m,   the   number   of    undulations   in 

the   ring.      When   m  =  ^,    the    values   of  L   and    N   will   be   at   their   maximum 

and  M=0.  If  we  determine  the  relation  between  S  and  R  in  this  case  so 
that  the  system  may  be  stable,  the  stability  of  the  system  for  every  other 
displacement  will  be  secured. 

8.  To  find  the  mass  which  must  be  given  to  the  central  body  in  order 
that  a  ring  of  satellites  may  permanently  revolve  round  it. 

We  have  seen  that  when  the  attraction  of  the  central  body  is  sufficiently 
great  compared  with  the  forces  arising  from  the  mutual  action  of  the  satellites, 
a  permanent  ring  is  possible.  Now  the  forces  between  the  satellites  depend  on 
the  manner  in  which  the  displacement  of  each  satellite  takes  place.  The  con- 
ception of  a  perfectly  arbitrary  displacement  of  all  the  satellites  may  be  rendered 
manageable  by  separating  it  into  a  number  of  partial  displacements  depending 
on  periodic  functions.  The  motions  arising  from  these  small  displacements  will 
take  place  independently,  so  that  we  have  to  consider  only  one  at  a  time. 

Of  all  these  displacements,  that  which  produces  the  greatest  disturbing 
forces    is    that   in  w^hich   consecutive   satellites   are   oppositely   displaced,    that  is, 

when   m  =  -,   for   then   the   nearest  satellites   are   displaced   so   as   to   increase   as 

z 

much  as  possible  the  effects  of  the  displacement  of  the  satellite  between  them. 
If  we  make  /x  a  large  quantity,  we  shall  have 

2™^<^  =  e;(l  +  3-'  +  5-  +  &c.)  =  ^.(l-0518). 
sm^  0  n'  ^  TT 

M=0,         N=2L,         J5r  very  small. 


IT 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  319 

Let   -  RL  =  X,  then  the  equation  of  motion  will  be 
/A 

n*-{S-x)n'  +  2x{'iS-x)=U=0 (23). 

The  conditions  of  tliis  equation  having  real  roots  are 

S>x  (24), 

(S-xY>^x{'iS-x)  (25). 

The  last  condition  gives  the  equation 

6:'-26*Sx  +  9ar>0, 

whence  S>2Q-U2x,    or>S<0-351a; (26). 

The  last  solution  is  inadmissible  because  S  must  be  greater  than  x,  so  that 

the  true  condition  is  »S>25*649a:, 

>  25-649  i  72^3 -5259, 

/X         IT 

S>-ASd2im'R  (27). 

So  that  if  there  were   100  satellites  in  the  ring,  then 

5>4352i2 
is   the    condition  which  must  be  fulfilled  in  order   that   the    motion   arising  from 
every  conceivable  displacement  may  be  periodic. 

If  this  condition  be  not  fulfilled,  and  if  S  be  not  sufiadent  to  render  the 
motion  perfectly  stable,  then  although  the  motion  depending  upon  long  undu- 
lations may  remain  stable,  the  short  undulations  wiill  increase  in  amplitude  till 
some  of  the  neighbouring  satellites  are  brought  into  collision. 

9.  To  determine  the  nature  of  the  motion  when  the  system  of  satellites 
is  of  small  mass  compared  with  the  central  body. 

The  equation  for  the  determination  of  n  is 

^        /x      ^  /x 

+  {Zoy-- R{2K+L)]~  RN -\R'M'=^0 (28). 

F'  r"  r" 

When  R  is  very  small  we  may  approximate  to  the  values  of  n  by  assuming 
that  two  of  them  are  nearly  ±  co,  and  that  the  other  two  are  small. 


320  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

If  we  put  n=  ±(0, 


dU 

dn 


=  ±2g>'  +  &c. 


Therefore  the  corrected  values  of  n  are 


n^±{<o  +  ^R(2K  +  L-.m)}  +  ^RM. 


(29). 


The   small   values  of  n  are   nearly  ±/3-i2iV^:  correcting  them  in   the 
way,  we  find  the  approximate  values 

n=±./3^EN^2~RM 


same 


(30). 


The  four  values  of  n  are  therefore 

1 


^1=  -<o-^-E{2K+L^iM-4N) 


RN-  —  RM 

fXCt) 


^z=+J^-RN-  —  RM 


(31), 


^4=  +o>+^--R(2K+L  +  iM-4N) 

and  the  complete   expression  for  p,  so  far  as  it  depends  on  terms  containing  ms, 
is  therefore  P  =  A,  cos  {ms  +  n^t  +  a^)-\-A^  cos  (ws  +  n^t  +  c^) 

+  A^co&(ms  +  nJ,  +  a^-{-A^coB{ms-\-nJ^  +  a^) (32), 

and   there   will   be   other  systems,  of  four  terms   each,   for  every   value   of  m  in 
the  expansion  of  the  original  disturbance. 

We  are  now  able  to  determine  the  value  of  o-  from  equations  (12),  (20),  by 
putting  /8  =  a,  and 


2<an  +  -  RM 


5=  — 


(33). 


n'  +  -RN 


ON    THE    STABILITY    OF    THE    MOTION     OF    SATURN'S    RINGS.  321 

So  that  for  every  term  of  p  of  the  form 

p  =  Acos  (ms  -{-111  + a)  (34), 

there  is  a  corresponding  term  in  a, 


2w7i  +  -  RM 


7t'  +  -RN 


A  sin  {ms-¥7it  + a) (35). 


10.  Let  us  now  fix  our  attention  on  the  motion  of  a  single  satellite, 
and  determine  its  motion  by  tracing  the  changes  of  p  and  a-  while  t  varies 
and  5  is  constant,  and  equal  to  the  value  of  s  corresponding  to  the  satellite 
in   question. 

We  must  recollect  that  p  and  a-  are  measured  outwards  and  forwards  from 
an  imaginary  point  revolving  at  distance  1  and  velocity  o,  so  that  the  motions 
we  consider  are  not  the  absolute  motions  of  the  satellite,  but  its  motions 
relative  to  a  point  fixed  in  a  revolving  plane.  This  being  understood,  we  may 
describe  the  motion  as  elliptic,  the  major  axis  being  in  the  tangential  direc- 
tion, and  the  ratio  of  the  axes  being  nearly  2  ^ ,  which  is  nearly  2  for  n,  and  n, 
and  is  very  large  for  n^  and  n^. 

The    time   of   revolution   is  — ,  or   if  we   take   a  revolution  of   the   ring   as 

the    unit   of    time,   the    time    of    a  revolution   of    the    satellite    about    its   mean 

...        .    it) 
position  IS  -  . 


The   direction   of  revolution   of   the   satellite   about  its   mean    position    is   in 
every  case  opposite  to  that  of  the  motion  of  the  ring. 

11.     The    absolute    motion   of   a   satellite    may    be    found    from    its    motion 
relative  to  the   ring  by  writing 

r=l+p  =  l+^cos  {ms  +  nt  +  a), 
d  =  (ot  +  s-{-<T  =  (ot  +  s-2  -Asm{ms-\-nt-\-a). 


322  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

When  n  is  nearly  equal  to  ±(0,  the  motion  of  each  satellite  in  space  is 
nearly  elliptic.  The  eccentricity  is  A,  the  longitude  at  epoch  s,  and  the  longi- 
tude when  at  the  greatest  distance  from  Saturn  is  for  the  negative  value  n^ 

-  —  R{2K+L-iM-4N)t  +  {m+l)s  +  a, 
and  for  the  positive  value  n^ 

-  —  R{2K+L  +  4M^4.N)t-{m+l)s-a. 

We  must  recollect  that  in  all  cases  the  quantity  within  brackets  is  negative, 
so  that  the  major  axis  of  the  ellipse  travels  forwards  in  both  cases.  The  chief 
difference  between  the  two  cases  lies  in  the  arrangement  of  the  major  axes  of 
the  ellipses  of  the  different  satellites.  In  the  first  case  as  we  pass  from  one 
satellite  to  the  next  in  front  the  axes  of  the  two  ellipses  lie  in  the  same 
order.  In  the  second  case  the  particle  in  front  has  its  major  axis  behind  that 
of  the  other.  In  the  cases  in  which  n  is  small  the  radius  vector  of  each 
satellite  increases  and  diminishes  during  a  periodic  time  of  several  revolutions. 
This  gives  rise  to  an  inequality,  in  which  the  tangential  displacement  far  exceeds 
the  radial,  as  in  the  case  of  the  annual  equation  of  the  Moon. 

12.  Let  us  next  examine  the  condition  of  the  ring  of  satellites  at  a  given 
instant.  We  must  therefore  fix  on  a  particular  value  of  t  and  trace  the  changes 
of  p  and  <r  for  different  values  of  s. 

From  the  expression  for  p  we  learn  that  the  satellites  form  a  wavy  line, 
which  is  furthest  from  the  centre  when  (ms  +  nt  +  a)  is  a  multiple  of  27r,  and 
nearest  to  the  centre  for  intermediate  values. 

From  the  expression  for  cr  we  learn  that  the  satellites  are  sometimes  in 
advance  and  sometimes  in  the  rear  of  their  mean  position,  so  that  there  are 
places  where  the  satellites  are  crowded  together,  and  others  where  they  are 
drawn  asunder.  When  n  is  positive,  ^  is  of  the  opposite  sign  to  A,  and  the 
crowding  of  the  satellites  takes  place  when  they  are  furthest  from  the  centre. 
When  n  is  negative,  the  satellites  are  separated  most  when  furthest  from  the 
centre,  and  crowded  together  when  they  approach  it. 

The  form  of  the  ring  at  any  instant  is  therefore  that  of  a  string  of  beads 
forming  a  re-entering  curve,  nearly  circular,  but  with  a  small  variation  of  distance 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  323 

from  the  centre  recurring  m  times,  and  forming  m  regular  waves  of  trans- 
vei-se  displacement  at  equal  intervals  round  the  circle.  Besides  these,  there  are 
waves  of  condensation  and  rarefaction,  the  effect  of  longitudinal  displacement. 
When  n  is  positive  the  points  of  greatest  distance  from  the  centre  are  points 
of  greatest  condensation,  and  when  n  is  negative  they  are  points  of  greatest 
rarefaction. 

13.  We  have  next  to  determine  the  velocity  with  which  these  waves  of 
disturbance  are  propagated  round  the  ring.  We  fixed  our  attention  on  a  par- 
ticular satellite  by  making  s  constant,  and  on  a  particular  instant  by  making  t 
constant,  and  thus  we  determined  the  motion  of  a  satellite  and  the  form  of  the 
ring.  We  must  now  fix  our  attention  on  a  phase  of  the  motion,  and  this  we 
do   by  making  p  or  a-  constant.      This  implies 

ms  +  nt  +  a  =  constant, 

ds  _      n 
dt~     m* 

So   that  the  particular  phase  of  the   disturbance  travels  round  the  ring  with  an 

angular  velocity  = relative   to   the   ring    itself.      Now   the    ring   is   revolving 

in  space  with  the  velocity  w,  so  that  the  angular  velocity  of  the  wave  in  space  is 

tj-  =  w (36). 

m 

Thus  each  satellite  moves  in  an  ellipse,  while  the  general  aspect  of  the 
ring  is  that  of  a  curve  of  m  waves  revolving  with  velocity  ct.  This,  however, 
is  only  the  part  of  the  whole  motion,  which  depends  on  a  single  term  of  the 
solution.  In  order  to  understand  the  general  solution  we  must  shew  how  to 
determine  the  whole  motion  from  the  state  of  the  ring  at  a  given  instant. 

14.  Given  the  position  and  motion  of  every  satellite  at  any  one  time,  to 
calculate  the  position  and  motion  of  every  satellite  at  any  other  time,  provided 
that  the  condition  of  stability  is  fulfilled. 

The  position  of  any  satellite  may  be  denoted  by  the  values  of  p  and  cr  for 
that  satellite,  and  its  velocity  and  direction  of  motion  are  then  indicated  by  the 

values  of  -r  and  -y-  at  the  g:iven  instant. 
dt  at 


324  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

These  four  quantities  may  have  for  each  satellite  any  four  arbitrary  values, 
as  the  position  and  motion  of  each  satellite  are  independent  of  the  rest,  at  the 
beginning  of  the  motion. 

Each  of  these  quantities  is  therefore  a  perfectly  arbitrary  ftmction  of  s,  the 
mean  angular  position  of  the  satellite  in  the  ring. 

But  any  function  of  s  from  s  =  0  to  s  =  27r,  however  arbitrary  or  discontinuous, 
can  be  expanded  in  a  series  of  terms  of  the  form  A  cos  (5  +  a)  +  A'  cos  (2s  +  a')  +  &c. 
See  §  3. 

Let  each  of  the  four  quantities  p,  -^ ,  a,  -j-  he  expressed  in  terms  of  such 

a  series,  and  let  the  terms  in  each  involving  ms  be 

p  =  Ecoa{'ms  +  e) (37), 

^^=Fcos(ins+f) (38). 

<T  =G  cos  (ms+g) (39), 

^  =  Hco3{ms  +  h) (40). 

These  are  the  parts  of  the  values  of  each  of  the  four  quantities  which  are 
capable  of  being  expressed  in  the  form  of  periodic  fimctions  of  ms.  It  is 
evident  that  the  eight  quantities  E,  F,  G,  H,  e,  f,  g,  h,  are  all  independent  and 
arbitrary. 

The  next  operation  is  to  tind  the  values  of  X,  M,  N,  belonging  to  disturb- 
ances in  the  ring  whose  index  is  m  [see  equation  (8)],  to  introduce  these 
values  into  equation  (28),  and  to  determine  the  four  values  of  n,  (ti,,  tIj,  1I3,  n^). 

This  being  done,  the  expression  for  p  is  that  given  in  equation  (32),  which 
contains  eight  arbitrary  quantities  (A,,  A^,  A3,  At,  «„  a^,  a^,  aj. 

Giving  t  its  original  value  in  this  expression,  and  equating  it  to  Eco3{7m-\-e), 
we  get  an  equation  which  is  equivalent  to  two.     For,  putting  7ns  =  0,  we  have 
^1  cos  Oi  +  .^2  cos  a,  +  -^3  cos  a,  +  ^^  cos  a^  =  -E'  cos  e (41). 

And  putting  ms=    ,  we  have  another  equation 

-4i  sin  Oi  +  ^j  sin  aj  +  ^3  sin  03  +  ^<  sin  a^  =  ^  sin  e (42). 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATUBN's    RINGS.  325 

Differentiating  (32)  with  respect  to  t,  we  get  two  other  equations 

-  A^n^  Bina-kc.-F cos/ (43), 

Aji^  cos  a  +  &c.=F  sin/ (44 ). 

Bearing  in  mind  that  B„  B^,  &c.  are  connected  with  A„  A^,  &c.  by  equa- 
tion (33),  and  that  B  is  therefore  proportional  to  A,  we  may  write  B  =  A^, 
where 

2o)n  +  -  RM 

P                   ^ 
P= 7 

H' 
^  being  thus  a  fiinction  of  n  and  a  known  quantity. 

The  value  of  <r  then  becomes  at  the  epoch 

<r  =  ^i)8i  sin  (m5  4- Oi) -I- &c.  =  Gcoa('ms-\-g), 
from  which  we  obtain  the  two  equations 

^^1  sin  Oi  -I-  &c.  =  6^  cos  g (45), 

^^iC0Sai  +  &c.  =  —Geing (46). 

Differentiating  with  respect  to  t,  we  get  the  remaining  equations 

A^jij^  cos  Oj  +  &c.  =  ^  cos  A (47), 

^^iniSinai-l-&c.  =  iZ'sinA (48). 

We  have  thus  found  eight  equations  to  determine  the  eight  quantities 
^1,  &c.  and  Oi,  &c.  To  solve  them,  we  may  take  the  four  in  which  -^iCosoi, 
&c.  occur,  and  treat  them  as  simple  equations,  so  as  to  find  ^iCosoj,  &c.  Then 
taking  those  in  which  ^isinoi,  &c.  occur,  and  determining  the  values  of  those 
quantities,  we  can  easily  deduce  the  value  of  A^  and  a,,   &c.  from  these. 

We  now  know  the  amplitude  and  phase  of  each  of  the  four  waves  whose 
index  is  m.  All  other  systems  of  waves  belonging  to  any  other  index  must 
be  treated  in  the  same  way,  and  since  the  original  disturbance,  however  arbitrary, 
can  be  broken  up  into  periodic  functions  of  the  form  of  equations  (37 — 40), 
our  solution  is  perfectly  general,  and  applicable  to  every  possible  disturbance  of 
a  ring  fulfilling  the  condition  of  stability  (27). 


326  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

15.  We  come  next  to  consider  the  effect  of  an  external  disturbing  force, 
due  either  to  the  irregularities  of  the  planet,  the  attraction  of  satellites,  or 
the  motion  of  waves  in  other  rings. 

All  disturbing  forces  of  this  kind  may  be  expressed  in  series  of  which  the 
general  term  is 

A  cos  {vt  +  ms  +  a), 

where  v  is  an  angular  velocity  and  m  a  whole  number. 

Let  P  cos  {ins  +  vt  +p)  be  the  central  part  of  the  force,  acting  inwards,  and 
Q  sin  (ms  +  vt  +  q)  the  tangential  part,  acting  forwards.  Let  p  =  A  cos  {tus  +  vt  +  a) 
and  a-  =  Bsm  (ms  +  vt-]-  fi),  be  the  terms  of  p  and  a  which  depend  on  the 
external  disturbing  force.  These  will  simply  be  added  to  the  terms  depending 
on  the  original  disturbance  which  we  have  already  investigated,  so  that  the 
complete  expressions  for  p  and  <t  will  be  as  general  as  before.  In  consequence 
of  the  additional  forces  and  displacements,  we  must  add  to  equations  (16)  and 
(17),  respectively,  the  following  terms: 

{Zar --R  (2K+  L)  +  v"]  A  cos  (m^-{-vt-\- a) 

+  (2q)V -\- -  RM)  B  COS  (ms  +  vt  +  f3)-P  cos  (ms  +  vt-hp)  =  0 (49). 

(2a)i;  4-  -  EM)  A  sin  (ms  +  vt  +  a) 

+  (v" +  - EN)  B  Bm(ms  +  vt  +  fi)-¥Q  sin  (ms  +  vt  +  q)  =  0 (50). 

Making  7ns  +  vt  =  0  in  the  first  equation   and  -  in  the  second, 

{S(o'--  E  (2K+L)  +  if}  A  cos  a  +  (2(ov  +  -E3f)  B  cos  fi-P  coap  =  0 (51). 

(2a>v  +  - EM)  A  cosa  +  (v'  +  -  EN) B COB  fi  +  Qcosq  =  0 (52). 


Then  if  we  put 
U'  =  v'-{oj'  +  -E(2K+L-N)}v'-A-EMv 


+  {Sa>'--E(2K+L)}-EN-\E'M' (53), 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  327 

we  shall  find  the  value  of  A  cos  a  and  B  coa  fi ; 

v'  +  -RN                 2cov-i-~RM 
A  cosa  = ft;        P  coa  p  + t4 Qcoaq (54). 

2(ov  4-  -  RM                y'  +  3<o'--R  {K+  L) 
Bcoafi= j^ Pcoap jp Qcoaq (55). 

Substituting  sines  for  cosines  in  equations  (51),  (52),  we  may  find  the 
values  of  A  sin  a  and  B  sin  ^. 

Now  U*  is  precisely  the  same  function  of  v  that  Z7  is  of  ?i,  so  that  if  u 
coincides  with  one  of  the  four  values  of  n,  U'  will  vanish,  the  coefiicients  A 
and  B  will  become  infinite,  and  the  ring  will  be  destroyed.  The  disturbing 
force  is  supposed  to   arise  from  a  revolving   body,  or  an   undulation  of  any  kind 

which    has    an    angular    velocity relatively    to    the   ring,   and   therefore   an 

absolute  angular  velocity  =  w . 

If  then  the  absolute  angular  velocity  of  the  disturbing  body  is  exactly  or 
nearly  equal  to  the  absolute  angular  velocity  of  any  of  the  free  waves  of  the 
ring,  that  wave  will  increase  till  the  ring  be  destroyed. 

The  velocities  of  the  free  waves  are  nearly 

l+i\     a>  +  i     /s-i^.V,  o>--     /s-i^iV^,    and  0)  fl-i) (56). 

When  the  angular  velocity  of  the  disturbing  body  is  greater  than  that  of 
the  first  wave,  between  those  of  the  second  and  third,  or  less  than  that  of 
the  fourth,  U'  is  positive.  When  it  is  between  the  first  and  second,  or  between 
the  third  and  fourth,    U'  is  negative. 

Let  us  now  simplify  our  conception  of  the  disturbance  by  attending  to  the 
central  force  only,  and  let  us  put  ^  =  0,  so  that  P  is  a  maximum  when  ms  +  vt 
is  a  multiple  of  27r.     We  find  in  this  case  a  =  0,  and  /8  =  0.     Also 

if+^-  RN 
^=—^P (57), 

2cjv +  -RM 
B= ^. P (58). 


328  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

When  U'  is  positive,  A  will  be  of  the  same  sign  as  P,  that  is,  the  parts 
of  the  ring  wlU  be  furthest  from  the  centre  where  the  disturbing  force  towards 
the  centre  is  greatest.     When   U'  is  negative,  the  contrary  will  be  the  case. 

When  V  is  positive,  B  will  be  of  the  opposite  sign  to  A,  and  the  parts 
of  the  ring  furthest  from  the  centre  will  be  most  crowded.  When  v  is  negative, 
the  contrary  will  be  the  case. 

Let  us  now  attend  only  to  the  tangential  force,  and  let  us  put  ^'  =  0.  We 
find  in  this  case  also  a  =  0,  )3  =  0, 

2(ov+-RM 

^= — tr — ^ (^^)' 

B= ^. Q (60). 

The  tangential  displacement  is  here  in  the  same  or  in  the  opposite  direc- 
tion to  the  tangential  force,  according  as  £/"'  is  negative  or  positive.  The 
crowding  of  sateUites  is  at  the  points  farthest  from  or  nearest  to  Saturn 
according  as  -y  is  positive  or  negative. 

16.  The  effect  of  any  disturbing  force  is  to  be  determined  in  the  following 
manner.  The  disturbing  force,  whether  radial  or  tangential,  acting  on  the  ring 
may  be  conceived  to  vary  from  one  satellite  to  another,  and  to  be  different  at 
different  times.     It  is  therefore  a  perfectly  arbitrary  function  of  s  and  t. 

Let  Fourier's  method  be  applied  to  the  general  disturbing  force  so  as  to 
divide  it  up  into  terms  depending  on  periodic  functions  of  s,  so  that  each  term 
is  of  the  form  F  (t)  cos  {ms  +  a),  where  the  function  of  i  is  still  perfectly  arbitrary. 

But  it  appears  from  the  general  theory  of  the  permanent  motions  of  the 
heavenly  bodies  that  they  may  all  be  expressed  by  periodic  functions  of  t 
arranged  in  series.  Let  vt  be  the  argument  of  one  of  these  terms,  then  the 
corresponding  term  of  the  disturbance  will  be  of  the  form 

P  cos  (ttis  +  vt  +  a). 

This  term  of  the  disturbing  force  indicates  an  alternately  positive  and 
negative    action,    disposed    in    m    waves    round  the  ring,   completing   its  period 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  329 

relatively    to    eaxih    particle   in   the   time    — ,   and   travelling   as   a   wave   among 

the  particles  with  an  angular  velocity ,   the  angular  velocity  relative  to  fixed 

space  being  of  course  oj  —  -  .     The    whole   disturbing  force    may    be   split   up  into 
terms  of  this  kind. 

17.  Each  of  these  elementary  disturbances  will  produce  its  own  wave  in 
the  ring,  independent  of  those  which  belong  to  the  ring  itself.  This  new  wave, 
due  to  external  disturbance,  and  following  different  laws  from  the  natural  waves 
of  the  rincy,  is  called  the  farced  wave.  The  angular  velocity  of  the  forced  wave 
is  the  same  as  that  of  the  disturbing  force,  and  its  maxima  and  minima  coin- 
cide with  those  of  the  force,  but  the  extent  of  the  disturbance  and  its  direction 
depend  on  the  comparative  velocities  of  the  forded  wave  and  the  four  natural 
waves. 

When  the  velocity  of  the  forced  wave  lies  between  the  velocities  of  the 
two  middle  free  waves,  or  is  greater  than  that  of  the  swiftest,  or  less  than 
that  of  the  slowest,  then  the  radial  displacement  due  to  a  radial  disturbing 
force  is  in  the  same  direction  as  the  force,  but  the  tangential  displacement 
due  to  a  tangential  disturbing  force  is  in  the  opposite  direction  to  the  force. 

The  radial  force  therefore  in  this  case  produces  a  positive  forced  wave,  and 
the  tangential  force  a  negative  forced  ivave. 

When  the  velocity  of  the  forced  wave  is  either  between  the  velocities  of 
the  first  and  second  free  waves,  or  between  those  of  the  third  and  fourth,  then 
the  radial  disturbance  produces  a  forced  wave  in  the  contrary  direction  to  that 
in  which  it  acts,  or  a  negative  wave,  and  the  tangential  force  produces  a  positive 
wave. 

The  coefficient  of  the  forced  wave  changes  sign  whenever  its  velocity  passes 
through  the  value  of  any  of  the  velocities  of  the  free  waves,  but  it  does  so 
by  becoming  infinite,  and  not  by  vanishing,  so  that  when  the  angular  velocity 
very  nearly  coincides  with  that  of  a  free  wave,  the  forced  wave  becomes  very 
great,  and  if  the  velocity  of  the  disturbing  force  were  made  exactly  equal  t-o 
that  of  a  free  wave,  the  coefficient  of  the  forced  wave  would  become  infinite. 
In  such  a  case  we  should  have  to  readjust  our  approximations,  and  to  find 
whether  such  a  coincidence  might  involve  a  physical  impossibility. 


330  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

The  forced  wave  which  we  have  just  investigated  is  that  which  would  main- 
tain itself  in  the  ring,  supposing  that  it  had  been  set  agoing  at  the  commence- 
ment of  the  motion.  It  is  in  fact  the  form  of  dynamical  equiUbrium  of  the 
ring  under  the  influence  of  the  given  forces.  In  order  to  find  the  actual  motion 
of  the  ring  we  must  combine  this  forced  wave  with  all  the  free  waves,  which 
go  on  independently  of  it,  and  in  this  way  the  solution  of  the  problem  becomes 
perfectly  complete,  and  we  can  determine  the  whole  motion  under  any  given 
initial  circumstances,  as  we  did  in  the  case  where  no  disturbing  force  acted. 

For  instance,  if  the  ring  were  perfectly  uniform  and  circular  at  the  instant 
when  the  disturbing  force  began  to  act,  we  should  have  to  combine  with  the 
constant  forced  wave  a  system  of  four  free  waves  so  disposed,  that  at  the  given 
epoch,  the  displacements  due  to  them  should  exactly  neutralize  those  due  to  the 
forced  wave.  By  the  combined  effect  of  these  four  free  waves  and  the  forced 
one  the  whole  motion  of  the  ring  would  be  accounted  for,  beginning  from  its 
undisturbed  state. 

The  disturbances  which  are  of  most  importance  in  the  theory  of  Saturn's 
rings  are  those  which  are  produced  in  one  ring  by  the  action  of  attractive 
forces  arising  from  waves  belonging  to  another  ring. 

The  effect  of  this  kind  of  action  is  to  produce  in  each  ring,  besides  its 
own  four  free  waves,  four  forced  waves  corresponding  to  the  free  waves  of  the 
other  ring.  There  will  thus  be  eight  waves  in  each  ring,  and  the  corresponding 
waves  in  the  two  rings  will  act  and  react  on  each  other,  so  that,  strictly  speak- 
ing, every  one  of  the  waves  will  be  in  some  measure  a  forced  wave,  although 
the  system  of  eight  waves  will  be  the  free  motion  of  the  two  rings  taken 
together.  The  theory  of  the  mutual  disturbance  and  combined  motion  of  two 
concentric  rings  of  satellites  requires  special  consideration. 

18.  On  the  motion  of  a  ring  of  satellites  when  the  conditions  of  stability 
are  not  fulfilled. 

We  have  hitherto  been  occupied  with  the  case  of  a  ring  of  satellites,  the 
stability  of  which  was  ensured  by  the  smaUness  of  mass  of  the  satellites  com- 
pared with  that  of  the  central  body.  We  have  seen  that  the  statically  unstable 
condition  of  each  satellite  between  its  two  immediate  neighbours  may  be  com- 
pensated by  the  dynamical  effect  of  its  revolution  round  the  planet,  and  a  planet 
of  sufiicient   mass   can   not   only  direct   the   motion   of    such   satellites   round    its 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S     RINGS.  331 

own  body,  but  can  likewise  exercise  an  influence  over  their  relations  to  each 
other,  so  as  to  overrule  their  natural  tendency  to  crowd  together,  and  distribute 
and  preserve  them  in  the  form  of  a  ring. 

We  have  traced  the  motion  of  each  satellite,  the  general  shape  of  the 
disturbed  ring,  and  the  motion  of  the  various  waves  of  disturbance  round  the 
ring,  and  determined  the  laws  both  of  the  natural  or  free  waves  of  the  ring, 
and  of  the  forced  waves,  due  to  extraneous  disturbing  forces. 

We  have  now  to  consider  the  cases  in  which  such  a  permanent  motion  of 
the  ring  is  impossible,  and  to  determine  the  mode  in  which  a  ring,  originally 
regular,  will  break  up,  in  the  different  cases  of  instability. 

The  equation  from  which  we  deduce  the  conditions  of  stability  is — 
U  =  n'-i(o'  +  -E(2K+L-N)\n'-4:(o-EMn 

+  hco'--R{2K  +  L)\-RN  -\r'M'  =  0. 

The  quantity,  which,  in  the  critical  cases,  determines  the  nature  of  the 
roots  of  this  equation,  is  N.  The  quantity  M  in  the  third  term  is  always 
small  compared  with  L  and  N  when  m  is  large,  that  is,  in  the  case  of  the 
dangerous  short  waves.  We  may  therefore  begin  our  study  of  the  critical  cases 
by  leaving  out  the  third  term.  The  equation  then  becomes  a  quadratic  in  n\ 
and  in  order  that  all  the  values  of  n  may  be  real,  both  values  of  n'  must  be 
real  and  positive. 

The  condition  of  the  values  of  n^  being  real  is 

oj*  +  co'-R{AK  +  2L-UN)  +  \b'{2K+L-\-NY>0 (61), 

which  shews  that  ay  must  either  be  about  14  times  at  least  smaller,  or  about  14 
times  at  least  greater,  than  quantities  like  -  RN. 


That  both  values  of  if  may  be  positive,  we  must  have 
co'  +  -R{2K  +  L-N)>0 


i3co''--R(2K-^L)\-RN>0 


(62). 


332  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    EINGS. 

We  must  therefore  take  the  larger  value  6£  oi\  and  also  add  the  condition 
that  N  be  positive. 

RN 

We  may  therefore  state  roughly,  that,  to  ensure  stability, ,  the  coefficient 

of  tangential  attraction,  must  lie  between  zero  and  -^oi\  If  the  quantity  be 
negative,  the  two  small  values  of  n  will  become  _pwre  impossible  quantities.  If 
it  exceed  ^oi\  all  the  values  of  n  will  take  the  form  of  mixed  impossible 
quantities. 

If  we   write  x  for  -  RN,  and  omit  the  other  disturbing  forces,  the  equation 

becomes  U=n*-{(o'-x)n'  +  Sco'x  =  0 (63), 

whence  n'  =  ^{co'-x)±^^/<o*-U(o'x  +  x'  (64). 

If  X  be  small,  two  of  the  values  of  n  are  nearly  ±<o,  and  the  others  are 
small  quantities,  real  when  x  is  positive  and  impossible  when  x  is  negative. 

2 

If  x  be  greater  than  {7-^IS)ar,  or  ^  nearly,  the  term  under  the  radical 
becomes  negative,  and  the  value  of  ?i  becomes 

n=  ±^^fjT2^  +  o}'-x±^/^-^'Jl2co'x-ajr  +  x  (65), 

where  one  of  the  terms  is  a  real  quantity,  and  the  other  impossible.  Every 
solution  may  be  put  under  the  form 

n=p±J^^q (66), 

where  ry  =  0  for  the  case  of  stability,  p  =  0  for  the  pure  impossible  roots,  and  p 
and  q  finite  for  the  mixed  roots. 

Let  us  now  adopt  this  general  solution  of  the  equation  for  n,  and  determine 
its  mechanical  significance  by  substituting  for  the  impossible  circular  functions 
their  equivalent  real  exponential  functions. 

Substituting  the  general  value  of  n  in  equations  (34),  (35), 

p  =  A[cos {ms +(p  +  'J^^q)t  +  a}  + cos {ms  +  ip- J -lq)t  + a}]  ...  (67), 


^^_^MP+±zlAsm{,ns  +  (p  +  ^^q)t  +  a}  ] 
(p  +  J-lqf  +  x 

_^MEpdIi^sm{ms+(p-sr^lq)t  +  a}    \ 
{p-'J  -IqY  +  x  J 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  333 

Introducing  the  exponential  notation,  these  values  become 

p  =  A(^^  +  €-''')co3(ms-{-pt  +  a)  (69), 

W r     2)  (^' +  r/ +  x)  (€«'  +  €-«')  sin  (771,5 +j9«  + a)  1 

We    have   now  obtained  a  solution  free  from  impossible  quantities,    and  applicable 
to  every  case. 

When  ^  =  0,  the  case  becomes  that  of  real  roots,  which  we  have  already 
discussed.  When  p  =  0,  we  have  the  case  of  pure  impossible  roots  arising  from 
the  negative  values  of  if.      The  solutions  corresponding  to  these  roots  are 

/3  =  ^  (e«' +  €-«')  cos  (m5  + a)  (71). 

o-=-^r^^^(€''-e-^0cos(m5  +  a) (72). 

The  part  of  the  coefficient  depending  on  e"''  diminishes  indefinitely  as  the 
time  increases,  and  produces  no  marked  effect.  The  other  part,  depending  on 
€^',  increases  in  a  geometrical  proportion  as  the  time  increases  arithmetically,  and 
so  breaks  up  the  ring.  In  the  case  of  x  being  a  small  negative  quantity,  q'  is 
nearly  3x,  so  that  the  coefficient  of  cr  becomes 

It  appears  therefore  that  the  motion  of  each  particle  is  either  outwards  and 
backwards  or  inwards  and  forwards,  but  that  the  tangential  part  of  the  motion 
greatly  exceeds  the  normal  part. 

It  may  seem  paradoxical  that  a  tangential  force,  acting  towards  a  position 
of  equilibrium,  should  produce  instability,  while  a  small  tangential  force  from  that 
position  ensures  stability,  but  it  is  easy  to  trace  the  destructive  tendency  of 
this  apparently  conservative  force. 

Suppose  a  particle  slightly  in  front  of  a  crowded  part  of  the  ring,  then 
if  X  is  negative  there  will  be  a  tangential  force  pushing  it  fonvards,  and  this 
force  will  cause  its  distance  from  the  planet  to  increase,  its  angular  velocity  U> 
diminish,  and  the  particle  itself  to  fall  back  on  the  crowded  part,  thereby 
increasing  the  irregularity  of  the  ring,  till  the  whole  ring  is  broken  up.  In 
the  same  way  it  may  be  shewn  that  a  particle  hehiiid  a  crowded  part  will  be 
pushed   into   it.      The    only  force  which  could   preserve    the    ring   from    the  effect 


334  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

of  tills  action,  is  one  which  would  prevent  the  particle  from  receding  from  the 
planet  under  the  influence  of  the  tangential  force,  or  at  least  prevent  the  dimi- 
nution of  angular  velocity.  The  transversal  force  of  attraction  of  the  ring  is  of 
this  kind,  and  acts  in  the  right  direction,  but  it  can  never  be  of  sufficient  magni- 
tude to  have  the  required  effect.  In  fact  the  thing  to  be  done  is  to  render  the 
last  term  of  the  equation  in  w  positive  when  N  is  negative,  which  requires 

fX 

and  this  condition  is  quite  inconsistent  with  any  constitution  of  the  ring  which 
fiilfils  the  other  condition  of  stability  which  we  shall  arrive  at  presently. 

We  may  observe  that  the  waves  belonging  to  the  two  real  values  of  n, 
±(D,  must  be  conceived  to  be  travelling  round  the  ring  during  the  whole  time 
of  its  breaking  up,  and  conducting  themselves  like  ordinary  waves,  till  the 
excessive  irregularities  of  the  ring  become  inconsistent  with  their  uniform  propa- 
gation. 

The  irregularities  which  depend  on  the  exponential  solutions  do  not  travel 
round  the  ring  by  propagation  among  the  sateUites,  but  remain  among  the  same 
satellites  which  first  began  to  move  irregularly. 

We  have  seen  the  fate  of  the  ring  when  x  is  negative.  When  x  is  small 
we  have  two  small  and  two  large  values  of  n,  which  indicate  regular  waves, 
as  we  have  already  shewn.  As  x  increases,  the  small  values  of  n  increase,  and 
the  large  values  diminish,  till  they  meet  and  form  a  pair  of  positive  and  a 
pair  of  negative  equal  roots,  having  values  nearly  +"68w.  When  x  becomes 
greater  than  about  -^(o",  then  all  the  values  of  n  become  impossible,  of  the 
form  ^j-F-n/  — Ig",  q  being  small  when  x  first  begins  to  exceed  its  limits,  and  p 
being  nearly   +  '6S(o. 

The  values  of  p  and  cr  indicate  periodic  inequalities  having  the  period  —  , 

but  increasing  in  amplitude  at  a  rate  depending  on  the  exponential  e''.  At  the 
beginning  of  the  motion  the  oscillations  of  the  particles  are  in  eUipses  as  in  the 
case  of  stability,  having  the  ratio  of  the  axes  about  1  in  the  normal  direction 
to  3  in  the  tangential  direction.  As  the  motion  continues,  these  ellipses  increase 
in  magnitude,  and  another  motion  depending  on  the  second  term  of  cr  is  com- 
bined with  the  former,  so  as  to  increase  the  ellipticity  of  the  oscillations  and  to 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  335 

turn  the  major  axis  into  an  inclined  position,  so  that  its  fore  end  points  a  little 
inwards,  and  its  hinder  end  a  little  outwards.  The  oscillations  of  each  particle 
round  its  mean  position  are  therefore  in  ellipses,  of  which  both  axes  increase 
continually  while  the  eccentricity  increases,  and  the  major  axis  becomes  sUghtly 
inclined  to  the  tangent,  and  this  goes  on  till  the  ring  is  destroyed.  In  the 
mean  time  the  irregularities  of  the  ring  do  not  remain  among  the  same  set  of 
particles  as  in  the  former  case,  but  travel  round  the  ring^  with  a  relative  angular 

velocity  -  ^^     Of  these  waves  there  are  four,  two  travelling  forwards  among  the 

satellites,  and  two  travelling  backwards.  One  of  each  of  these  pairs  depends 
on  a  negative  value  of  q,  and  consists  of  a  wave  whose  amplitude  continually 
decreases.  The  other  depends  on  a  positive  value  of  q,  and  is  the  destructive 
wave  whose  character  we  have  just  described. 

19.  We  have  taken  the  case  of  a  ring  composed  of  equal  satellites,  as 
that  with  which  we  may  compare  other  cases  in  which  the  ring  is  constructed 
of  loose  materials  diiferently  arranged. 

In  the  first  place  let  us  consider  what  will  be  the  conditions  of  a  ring 
composed  of  satellites  of  unequal  mass.  We  shall  find  that  the  motion  is  of 
the  same  kind  as  when  the  satellites  are  equal. 

For  by  arranging  the  satellites  so  that  the  smaller  satellites  are  closer 
together  than  the  larger  ones,  we  may  form  a  ring  which  will  revolve  uni- 
formly about  Saturn,  the  resultant  force  on  each  satellite  being  just  sufficient 
to  keep  it  in  its  orbit. 

To  determine  the  stability  of  this  kind  of  motion,  we  must  calculate  the 
disturbing  forces  due  to  any  given  displacement  of  the  ring.  This  calculation 
will  be  more  complicated  than  in  the  former  case,  but  will  lead  to  results  of 
the  same  general  character.  Placing  these  forces  in  the  equations  of  motion, 
we  shall  find  a  solution  of  the  same  general  character  as  in  the  former  case, 
only  instead  of  regular  waves  of  displacement  travelling  round  the  ring,  each 
wave  will  be  split  and  reflected  when  it  comes  to  irregularities  in  the  chain  of 
satellites.  But  if  the  condition  of  stability  for  every  kind  of  wave  be  fulfilled, 
the  motion  of  each  satellite  will  consist  of  small  oscillations  about  its  position 
of  dynamical  equilibrium,  and  thus,  on  the  whole,  the  ring  will  of  itself  assume 
the  arrangement  necessary  for  the  continuance  of  its  motion,  if  it  be  originally 
in  a  state  not  very  different  from  that  of  equilibrium. 


336  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN'S    RINGS. 

20.  We  now  pass  to  the  case  of  a  ring  of  an  entirely  different  construc- 
tion. It  is  possible  to  conceive  of  a  quantity  of  matter,  either  solid  or  liquid, 
not  collected  into  a  continuous  mass,  but  scattered  thinly  over  a  great  extent 
of  space,  and  having  its  motion  regulated  by  the  gravitation  of  its  parts  to 
each  other,  or  towards  some  dominant  body.  A  shower  of  rain,  hail,  or  cinders 
is  a  familiar  illustration  of  a  number  of  unconnected  particles  in  motion;  the 
visible  stars,  the  milky  way,  and  the  resolved  nebula?,  give  us  instances  of  a 
similar  scattering  of  bodies  on  a  larger  scale.  In  the  terrestrial  instances  we 
see  the  motion  plainly,  but  it  is  governed  by  the  attraction  of  the  earth,  and 
retarded  by  the  resistance  of  the  air,  so  that  the  mutual  attraction  of  the 
parts  is  completely  masked.  In  the  celestial  cases  the  distances  are  so  enor- 
mous, and  the  time  during  which  they  have  been  observed  so  short,  that  we 
can  perceive  no  motion  at  all.  StiU  we  are  perfectly  able  to  conceive  of  a 
collection  of  particles  of  small  size  compared  with  the  distances  between  them, 
acting  upon  one  another  only  by  the  attraction  of  gravitation,  and  revolving 
round  a  central  body.  The  average  density  of  such  a  system  may  be  smaller 
than  that  of  the  rarest  gas,  while  the  particles  themselves  may  be  of  great 
density ;  and  the  appearance  from  a  distance  will  be  that  of  a  cloud  of  vapour, 
with  this  difference,  that  as  the  space  between  the  particles  is  empty,  the  rays 
of  light  will  pass  through  the  system  without  being  refracted,  as  they  would 
have  been  if  the  system  had  been  gaseous. 

Such  a  system  will  have  an  average  density  which  may  be  greater  in  some 
places  than  others.  The  resultant  attraction  wiU  be  towards  places  of  greater 
average  density,  and  thus  the  density  of  those  places  wiU  be  increased  so  as 
to  increase  the  irregularities  of  density.  The  system  will  therefore  be  statically 
unstable,  and  nothing  but  motion  of  some  kind  can  prevent  the  particles  from 
forming  agglomerations,  and  these  uniting,  till  all  are  reduced  to  one  solid 
mass. 

We  have  already  seen  how  dynamical  stability  can  exist  where  there  is 
statical  instability  in  the  case  of  a  row  of  particles  revolving  round  a  central 
body.  Let  us  now  conceive  a  cloud  of  particles  forming  a  ring  of  nearly  uni- 
form density  revolving  about  a  central  body.  There  will  be  a  primary  effect  of 
inequalities  in  density  tending  to  draw  particles  towards  the  denser  parts  of  the 
ring,  and  this  will  ehcit  a  secondary  effect,  due  to  the  motion  of  revolution, 
tending  in  the  contrary  direction,  so  as  to  restore  the  rings  to  uniformity.     The 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  337 

relative  magnitude  of  these  two  opposing  forces  determines  the  destruction  or 
preservation  of  the  ring. 

To  calculate  these  effects  we  must  begin  with  the  statical  problem : — To 
determine  the  forces  arising  from  the  given  displacements  of  the  ring. 

The  longitudinal  force  arising  from  longitudinal  displacements  is  that  which 
has  most  effect  in  determining  the  stability  of  the  ring.  In  order  to  estimate  ita 
limiting  value  we  shall  solve  a  problem  of  a  simpler  form. 

21.  An  infinite  mass,  originally  of  uniform  density  Tc,  has  its  particles 
displaced  by  a  quantity  f  parallel  to  the  axis  of  x,  so  that  ^  =  AcQ^mx,  to 
determine  the  attraction  on  each  particle  due  to  this  displacement. 

The  density  at  any  point  will  differ  from  the  original  density  by  a  quantity 
k' ,  so  that 

{k  +  k')  (dx  +  d^)  =  kdx (73), 

k'=  —k-r-  =  Akm  sin  mx (74). 

The  potential  at  any  point  will  be  V+V,  where  V  is  the  original  potential, 
and   F'  depends  on  the  displacement  only,  so  that 

dT     d'V     d'V     ^    ,,     ^  ,^,, 

^+-5^  +  ^-  +  ^'^^=^ (^^)- 

Now   V  is  a  function  of  x  only,  and  therefore, 

V  =  AirAk  —sinmx (76), 

and  the  longitudinal  force  is  found  by  differentiating   V  with  respect  to  x. 

dV 
X=    -,—  =  ink  A  cos  mx  = 'ink^ (77). 

Now  let  us  suppose  this  mass  not  of  infinite  extent,  but  of  finite  section 
parallel  to  the  plane  of  yz.  This  change  amounts  to  cutting  off  all  portions 
of  the  mass  beyond  a  certain  boundary.  Now  the  effect  of  the  portion  so  cut 
off  upon  the  longitudinal  force  depends  on  the  value  of  m.  When  m  is  large, 
so  that  the  wave-length  is  small,  the  effect  of  the  external  portion  is  insensible, 
so  that  the  longitudinal  force  due  to  short  waves  is  not  diminished  by  cutting 
off  a  great  portion  of  the  mass. 


338  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

22.  Applying  this   result   to   the   case   of  a   ring,  and  putting  s  for  x,  and 

a-  for  $  we  have 

cr  =  ^  cos  ms,  and  T=  AttJcA  cos  ms, 

so  that  -RN=4:Trk, 

when  on  is  very  large,  and  this  is  the  greatest  value  of  N. 

The  value  of  L  has  little  effect  on  the  condition  of  stability.  If  L  and 
M  are  both  neglected,  that  condition  is 

(o'>27-S5e  (2nk) (78), 

and   if  L  be   as   much  as  ^N,  then 

o>^>25-649  (27rk) (79), 

so  that  it  is  not  important  whether  we  calculate  the  value  of  L  or  not. 

The  condition  of  stability  is,  that  the  average  density  must  not  exceed  a 
certain  value.  Let  us  ascertain  the  relation  between  the  maximum  density  of 
the  ring  and  that  of  the  planet. 

Let  h  be  the  radius  of  the  planet,  that  of  the  ring  being  unity,  then  the 
mass  of  Saturn  is  ^Trh'k'  =  o)"'  if  k'  be  the  density  of  the  planet.  If  we  assume 
that  the  radius  of  the  ring  is  twice  that  of  the  planet,  as  Laplace  has  done, 
then  h  =  ^  and 

1  =  334-2  to  307-7 (80), 

so  that  the  density  of  the  ring  cannot  exceed  3^  of  that  of  the  planet.  Now 
Laplace  has  shewn  that  if  the  outer  and  inner  parts  of  the  ring  have  the  same 
angular  velocity,  the  ring  will  not  hold  together  if  the  ratio  of  the  density  of 
the  planet  to  that  of  the  ring  exceeds  1-3,  so  that  in  the  first  place,  our  ring 
cannot  have  uniform  angular  velocity,  and  in  the  second  place,  Laplace's  ring 
cannot  preserve  its  form,  if  it  is  composed  of  loose  materials  acting  on  each 
other  only  by  the  attraction  of  gravitation,  and  moving  with  the  same  angular 
velocity  throughout. 

23.  On  the  forces  arising  from  inequalities  of  thickness  in  a  thin  stratum 
of  fluid  of  indefinite  extent. 

The  forces  which  act  on  any  portion  of  a  continuous  fluid  are  of  two  kinds, 
the  pressures  of  contiguous  portions  of  fluid,  and  the  attractions  of  all  portions  of 
the  fluid  whether  near  or  distant.     In  the  case   of  a  thin  stratum  of  fluid,  not 


ON    THE    STABILITY    OF    THE    MOTION     OF    SATURN's     RINGS.  339 

acted  on  by  any  external  forces,  the  pressures  are  due  mainly  to  the  component 
of  the  attraction  which  is  perpendicular  to  the  plane  of  the  stratum.  It  is 
easy  to  shew  that  a  fluid  acted  on  by  such  a  force  will  tend  to  assume  a 
position  of  equilibrium,  in  which  its  free  surface  is  plane  ;  and  that  any  irregu- 
larities will  tend  to  equalise  themselves,  so  that  the  plane  surface  will  be  one 
of  stable  equilibrium. 

It  is  also  evident,  that  if  we  consider  only  that  part  of  the  attraction 
which  is  parallel  to  the  plane  of  the  stratum,  we  shall  find  it  always  directed 
towards  the  thicker  parts,  so  that  the  effect  of  this  force  is  to  draw  the  fluid 
from  thinner  to  thicker  parts,  and  so  to  increase  irregularities  and  destroy 
equilibrium. 

The  normal  attraction  therefore  tends  to  preserve  the  stability  of  equilibrium, 
while  the  tangential  attraction  tends  to  render  equilibrium  unstable. 

According  to  the  nature  of  the  irregularities  one  or  other  of  these  forces 
will  prevail,  so  that  if  the  extent  of  the  irregularities  is  small,  the  normal 
forces  will  ensure  stability,  while,  if  the  inequaUties  cover  much  space,  the 
tangential  forces  will  render  equilibrium  unstable,  and  break  up  the  stratum  into 
beads. 

To  fix  our  ideas,  let  us  conceive  the  irregularities  of  the  stratum  split  up 
into  the  form  of  a  number  of  systems  of  waves  superposed  on  one  another, 
then,  by  what  we  have  just  said,  it  appears,  that  very  short  waves  will  disap- 
pear of  themselves,  and  be  consistent  with  stability,  while  very  long  waves  will 
tend  to  increase  in  height,  and  will  destroy  the  form  of  the  stratum. 

In  order  to  determine  the  law  according  to  which  these  opposite  effects 
take  place,  we  must  subject  the  case  to  mathematical  investigation. 

Let  us  suppose  the  fluid  incompressible,  and  of  the  density  k,  and  let  it 
be  originally  contained  between  two  parallel  planes,  at  distances  +c  and  —  c 
from  that  of  (xy),  and  extending  to  infinity.  Let  us  next  conceive  a  series  of 
imaginary  planes,  parallel  to  the  plane  of  {ijz),  to  be  plunged  into  the  fluid 
stratum  at  infinitesimal  distances  from  one  another,  so  as  to  divide  the  fluid 
into  imaginary  slices  perpendicular  to  the  plane  of  the  stratum. 

Next  let  these  planes  be  displaced  parallel  to  the  axis  of  x  according  to  this 
law — that  if  x  be  the  original  distance  of  the  plane  from  the  origin,  and  ^  its 
displacement  in  the  direction  of  x, 

i=A  cosmx (81). 


340  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

According  to  this  law  of  displacement,  certain  alterations  will  take  place  in 
the  distances  between  consecutive  planes ;  but  since  the  fluid  is  incompressible, 
and  of  indefinite  extent  in  the  direction  of  y,  the  change  of  dimension  must 
occur  in  the  direction  of  z.  The  original  thickness  of  the  stratum  was  2c.  Let 
its  thickness  at  any  point  after  displacement  be  2c +  2^,  then  we  must  have 


.+i)=2^ («2)' 

1=  — c  -r-=cmA  sinwa; (83). 


(2c +  20  (l 


Let  us  assume  that  the  increase  of  thickness  2^  is  due  to  an  increase  of  C, 
at  each  surface  ;  this  is  necessary  for  the  equilibrium  of  the  fluid  between  the 
imaginary  planes. 

We  have   now  produced  artificially,   by  means  of  these  planes,  a  system  of 

waves   of  longitudinal   displacement   whose   length   is    —   and  amplitude  A ;   and 

we  have  found  that  this   has    produced  a  system  of  waves   of  normal   displace- 
ment  on  each   surface,   having  the   same  length,  with  a  height  =cmA. 

In  order  to  determine  the  forces  arising  from  these  displacements,  we  must, 
in  the  first  place,  determine  the  potential  function  at  any  point  of  space,  and 
this  depends  partly  on  the  state  of  the  fluid  before  displacement,  and  partly 
on  the  displacement  itself     We  have,  in  all  cases — 

d'V     d'V     d'V 
^^+^  +  ^=-^^^ («^)- 

Within  the  fluid,  p  =  k;  beyond  it,  p  =  0. 
Before  displacement,  the  equation  is  reduced  to 

d^'  =  -'-p («^)- 

Instead  of  assuming  F=0  at  infinity,  we  shall  assume  F=0  at  the  origin, 
and  since  in  this  case  all  is  symmetrical,  we  have 

within  the  fluid  F,  =  -  2nkz'  -,    ^  =  -  inJcz 


at  the  bounding  planes  F=  —  iirkc^ ;    ->-  =  T  47r^c 

beyond  them  V,  =  27r^c  ( +  2z  ±  c) ;    -y-  =  =F  ^nkc 


.(86); 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S     RINGS.  341 

the   upper   sign   being   understood   to   refer   to    the   boundary  at  distance  +c,  and 
the  lower  to  the  boundary  at  distance  —  c  from  the  origin. 

Having  ascertained  the  potential  of  the  undisturbed  stratum,  we  find  that 
of  the  disturbance  by  calculating  the  effect  of  a  stratum  of  density  k  and 
thickness  t„  spread  over  each  surface  according  to  the  law  of  thickness  already 
found.  By  supposing  the  coeJB&cient  A  small  enough,  (as  we  may  do  in  calcu- 
lating the  displacements  on  which  stabiUty  depends),  we  may  diminish  the 
absolute  thickness  indefinitely,  and  reduce  the  case  to  that  of  a  mere  "  super- 
ficial density,"  such  as  is  treated  of  in  the  theory  of  electricity.  We  have  here, 
too,  to  regard  some  parts  as  of  negative  density  ;  but  we  must  recollect  that  we 
are  dealing  with  the  difference  between  a  disturbed  and  an  undisturbed  system, 
which  may  be  positive  or  negative,  though  no  real   mass  can  be  negative. 

Let  us  for  an  instant  conceive  only  one  of  these  surfaces  to  exist,  and  let 
us  transfer  the  origin  to  it.     Then  the  law  of  thickness  is 

l,  =  mcABm.'mx (83), 

and   we   know   that   the   normal   component   of  attraction   at   the   surface    is   the 
same  as  if  the  thickness  had  been  uniform  throughout,  so  that 

on  the  positive  side  of  the  surface. 
Also,  the  solution  of  the  equation 

d'V   dyv_ 

dx"  "^  dz'  ~    ' 

consists  of  a  series  of  terms  of  the  form  Ce'"  sin  ix. 

Of  these  the  only  one  with  which  we  have  to  do  is  that  in  which  i=  —m. 
Applying  the  condition  as  to  the  normal  force  at  the  surface,  we  get 

V=2'irkce''^Asmmx (87), 

for  the  potential  on  the  positive  side  of  the  surface,  and 

V=27rkce'^ABm7nx (88), 

on  the  negative  side. 


[ (89) 


342  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

Calculating  the  potentials  of  a  pair  of  such  surfaces  at  distances  +c  and  —c 
from  the  plane  of  xy,  and  calling  V  the  sum  of  their  potentials,  we  have  for 
the  space  between  these  planes 

F/  =  2TrkcA  sin  mxe"""  (e"^  +  e-*^) 
beyond  them  F/  =  27rZ;c^  sinma!;e^"^(e'^  +  e~'^) 

the  upper   or  lower  sign   of  the  index  being  taken  according  as  z  is  positive  or 
negative. 

These  potentials  must  be  added  to  those  formerly  obtained,  to  get  the 
potential  at  any  point  after  displacement. 

We  have  next  to  calculate  the  pressure  of  the  fluid  at  any  point,  on  the 
supposition  that  the  imaginary  planes  protect  each  shce  of  the  fluid  from  the 
pressure  of  the  adjacent  sHces,  so  that  it  is  in  equilibrium  under  the  action  of 
the  forces  of  attraction,  and  the  pressure  of  these  planes  on  each  side.  Now 
in  a  fluid  of  density  h,  in  equilibrium  under  forces  whose  potential  is  V,  we 
have  always — 

so  that  if  we  know  that  the  value  of  p  is  2\  where  that  of   F  is    F^,  then  at 

any  other  point 

jD=^„  +  ^(F-F„). 

Now,  at  the  free  surface  of  the  fluid,  ]p  =  0,  and  the  distance  from  the 
free  surface  of  the  disturbed  fluid  to  the  plane  of  the  original  surface  is  ^,  a 
small  quantity.  The  attraction  which  acts  on  this  stratum  of  fluid  is,  in  the 
first  place,  that  of  the  undisturbed  stratum,  and  this  is  equal  to  A^irkc,  towards 
that  stratum.  The  pressure  due  to  this  cause  at  the  level  of  the  original 
surface  will  be  AnJifcC,  and  the  pressure  arising  from  the  attractive  forces  due 
to  the  displacements  upon  this  thin  layer  of  fluid,  will  be  small  quantities  of 
the  second  order,  which  we  neglect.     We  thus  find  the  pressure  when  z  =  c  to  be, 

Pa  =  AvJc^c^mA  sin  mx. 

The  potential  of  the   undisturbed  mass  when  z  =  c  is 

V,=  -2TTkc\ 
and  the  potential  of  the  disturbance  itself  for  the   same  value  of  z,  is 

F;  =  2TrkcA  sin  mx  (1  +  e""^). 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  343 

So  that  we  find  the  general  value  of  jp  at  any  other  point  to  be 

^  =  27r^^  (c' -  z')  +  27r/:'c^  sin  ?7ia;  {2c»i  -  1  -  €- ^"^  +  e"^  (e"- +  e""^)}  . . .  (90). 

This  expression  gives  the  pressure  of  the  fluid  at  any  point,  as  depending 
on  the  state  of  constraint  produced  by  the  displacement  of  the  imaginary  planes. 
The  accelerating  effect  of  these  pressures  on  any  particle,  if  it  were  allowed  to 
move  parallel  to  x,  instead  of  being  confined  by  the  planes,  would  be 

_1  dp 
k  dx' 

The  accelerating  effect  of  the  attractions  in  the  same  direction  is 

dV 
dx' 

so  that  the  whole  acceleration  parallel  to  cc  is 

X=  -lirkmcA  cos  7nx  {2mc - e''^ -  I)  (91). 

It  is  to  be  observed,  that  this  quantity  is  independent  of  z,  so  that  every 
particle  in  the  slice,  by  the  combined  effect  of  pressure  and  attraction,  is  urged 
with  the  same  force,  and,  if  the  imaginary  planes  were  removed,  each  slice 
would  move  parallel  to  itself  without  distortion,  as  long  as  the  absolute  dis- 
placements remained  small.  We  have  now  to  consider  the  direction  of  the 
resultant  force  X,  and  its  changes  of  magnitude. 

We  must  remember  that  the  original  displacement  is  A  cos  7nx,  if  therefore 
(2mo-e~"^— 1)  be  positive,  X  will  be  opposed  to  the  displacement,  and  the 
equilibrium  will  be  stable,  whereas  if  that  quantity  be  negative,  X  will  act 
along  with  the  displacement  and  increase  it,  and  so  constitute  an  unstable 
condition. 

It  may  be  seen  that  large  values  of  nic  give  positive  results  and  small 
ones  negative.      The  sign  changes  when 

2mc  =  l'lA7 (92), 

which   corresponds   to   a  wave-length 

\  =  2c^^^  =  2c{5'i7l) (93). 

The  length  of  the  complete  wave  in  the  critical  case  is  5*471  times  the 
thickness  of  the  stratum.  Waves  shorter  than  this  are  stable,  longer  waves 
are  unstable. 


344  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

The  quantity  2mc{2mc-e-^-l), 

has  a  minimum  when  2mc  =  '607 (94), 

and  the  wave-length  is  10 '3 5 3  times  the  thickness  of  the  stratum. 

In  this  case  2mc  (2mc-e-^"^- 1)=  - '509 (95), 

and  X='5097rMcosmx  (96). 

24.  Let  us  now  conceive  that  the  stratum  of  fluid,  instead  of  being  infinite 
in  extent,  is  limited  in  breadth  to  about  100  times  the  thickness.  The  pressures 
and  attractions  will  not  be  much  altered  by  this  removal  of  a  distant  part  of 
the  stratum.  Let  us  also  suppose  that  this  thin  but  broad  strip  is  bent  round 
in  its  own  plane  into  a  circular  ring  whose  radius  is  more  than  ten  times  the 
breadth  of  the  strip,  and  that  the  waves,  instead  of  being  exactly  parallel  to 
each  other,  have  their  ridges  in  the  direction  of  radii  of  the  ring.  We  shall 
then  have  transformed  our  stratum  into  one  of  Saturn's  Kings,  if  we  suppose 
those  rings  to  be  liquid,  and  that  a  considerable  breadth  of  the  ring  has  the 
same  angular  velocity. 

Let  us  now  investigate  the  conditions  of  stability  by  putting 
x=  - 27rkmc  (2mc - e"^ -  1) 

into   the   equation    for   n.      We    know   that   x    must   lie   between   0   and  ^^  to 

ensure  stabihty.  Now  the  greatest  value  of  x  in  the  fluid  stratum  is  -50917^-. 
Taking  Laplace's  ratio  of  the  diameter  of  the  ring  to  that  of  the  planet,  this 
gives  42-5  as  the  minimum  value  of  the  density  of  the  planet  divided  by  that 
of  the  fluid  of  the  ring. 

Now  Laplace  has  shewn  that  any  value  of  this  ratio  greater  than  1-3  is 
inconsistent  with  the  rotation  of  any  considerable  breadth  of  the  fluid  at  the 
same  angular  velocity,  so  that  our  hypothesis  of  a  broad  ring  with  uniform 
velocity  is  untenable. 

But  the  stabihty  of  such  a  ring  is  impossible  for  another  reason,  namely, 
that  for  waves  in  which  2mc>  1-147,  x  is  negative,  and  the  ring  will  be  destroyed 
by  these  short  waves  in  the  manner  described  at  page  (333). 

When  the  fluid  ring  is  treated,  not  as  a  broad  strip,  but  as  a  filament  of 
circular   or  elliptic   section,  the   mathematical  difiSculties  are  very  much  increased. 


ON    THE    STABILITY    OF    THE     MOTION     OF    SATURN's    RINGS.  345 

but  it  may  be  shown  that  in  this  case  also  there  will  be  a  maximum  value 
of  X,  which  will  require  the  density  of  the  planet  to  be  several  times  that  of 
the  ring,  and  that  in  all  cases  short  waves  will  give  rise  to  negative  values 
of  X,  inconsistent  with  the  stability  of  the  rmg. 

It  appears,  therefore,  that  a  ring  composed  of  a  continuous  liquid  mass 
cannot  revolve  about  a  central  body  without  being  broken  up,  but  that  the 
parts  of  such  a  broken  ring  may,  under  certain  conditions,  form  a  permanent 
ring  of  satellites. 

On   the   Mutual   Perturbations   of   Two   Rings. 

25.  We  shall  assume  that  the  difference  of  the  mean  radii  of  the  rings 
is  small  compared  with  the  radii  themselves,  but  large  compared  with  the 
distance  of  consecutive  satellites  of  the  same  ring.  We  shall  also  assume  that 
each  ring  separately  satisfies  the  conditions  of  stability. 

We  have  seen  that  the  effect  of  a  disturbing  force  on  a  ring  is  to  produce 
a  series  of  waves  whose  number  and  period  correspond  with  those  of  the  dis- 
turbing force  which  produces  them,  so  that  we  have  only  to  calculate  the 
coefficient  belonging  to  the  wave  from  that  of  the  disturbing  force. 

Hence  in  investigating  the  simultaneous  motions  of  two  rings,  we  may 
assume  that  the  mutually  disturbing  waves  travel  with  the  same  absolute 
angular  velocity,  and  that  a  maximum  in  one  corresponds  either  to  a  maximum 
or  a  minimum  of  the  other,  according  as  the  coefficients  have  the  same  or 
opposite  signs. 

Since  the  motions  of  the  particles  of  each  ring  are  affected  by  the  disturbance 
of  the  other  ring,  as  well  as  of  that  to  which  they  belong,  the  equations  of 
motion  of  the  two  rings  will  be  involved  in  each  other,  and  the  final  equation 
for  determining  the  wave-velocity  will  have  eight  roots  instead  of  four.  But  as 
each  of  the  rings  has  four  free  waves,  we  may  suppose  these  to  originate  forced 
waves  in  the  other  ring,  so  that  we  may  consider  the  eight  waves  of  each  ring 
as  consisting  of  four  free  waves  and  four  forced  ones. 

In  strictness,  however,  the  wave- velocity  of  the  "free"  waves  will  be 
affected  by  the  existence  of  the  forced  waves  which  they  produce  in  the  other 
ring,  so  that  none  of  the  waves  are  really  "  free "  in  either  ring  independently, 
though  the  whole  motion  of  the  system  of  two  rings  as  a  whole  is  free. 


346  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

We  shall  find,  however,  that  it  is  best  to  consider  the  waves  first  as  free, 
and  then  to  determine  the  reaction  of  the  other  ring  upon  them,  which  is  such 
as  to  alter  the  wave-velocity  of  both,  as  we  shall  see. 

The  forces  due  to  the  second  ring  may  be  separated  into  three  parts. 

1st.     The  constant  attraction  when  both  rings  are  at  rest. 

2nd.  The  variation  of  the  attraction  on  the  first  ring,  due  to  its  own 
disturbances. 

3rd.  The  variation  of  the  attraction  due  to  the  disturbances  of  the  second 
ring. 

The  first  of  these  affects  only  the  angular  velocity.  The  second  affects  the 
waves  of  each  ring  independently,  and  the  mutual  action  of  the  waves  depends 
entirely  on  the  third  class  of  forces. 

26.     To  deteivnine  the  attractions  between  two  rings. 

Let  R  and  a  be  the  mass  and  radius  of  the  exterior  ring,  R  and  a'  those 
of  the  interior,  and  let  all  quantities  belonging  to  the  interior  ring  be  marked 
with  accented  letters.     (Fig.  5.) 

1st.     Attraction  between  the  rings  when  at  rest. 

Since  the  rings  are  at  a  distance  small  compared  with  their  radii,  we  may 
calculate  the  attraction  on  a  particle  of  the  first  ring  as  if  the  second  were  an 
infinite  straight  line  at  distance  a'  — a  from  the  first. 

7?' 

The  mass  of  unit  of  length  of  the  second  ring  is  - — > ,  and  the  accelerating 

effect  of  the  attraction  of  such  a  filament  on  an  element  of  the  first  ring  is 

TV 

— —, 7\  inwards  (97). 

na  [a  —  a)  ^ 

The  attraction  of  the  first  ring  on  the  second  may  be  found  by  transposing 
accented  and  unaccented  letters. 

In   consequence   of  these   forces,    the  outer  ring  will   revolve  faster,  and  the 

inner   ring   slower   than   would   otherwise  be   the   case.      These   forces   enter   into 

the   constant    terms   of    the   equations   of  motion,    and    may    be   included   in   the 
value  of  K. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  347 

2nd.     Variation  due  to  disturbance  of  first  ring. 

If  we  put  a(l+p)  for  a  in  the  last  expression,  we  get  the  attraction 
when  the  first  ring  is  displaced.     The  part  depending  on  p  is 

r-, TT,  P  inwards  (98). 

Tra  [a-ay  '^ 

This  is  the  only  variation  of  force  arising  from  the  displacement  of  the 
first  ring.     It  affects  the  value  of  X  in  the  equations  of  motion. 

3rd.      Variation  due  to  waves  in  the  second  ling. 

On  account  of  the  waves,  the  second  ring  varies  in  distance  from  the 
first,  and  also  in  mass  of  unit  of  length,  and  each  of  these  alterations  produces 
variations  both  in  the  radial  and  tangential  force,  so  that  there  are  four  things 
to  be  calculated : 

1st.  Radial  force  due  to  radial  displacement. 

2nd,  Radial  force  due  to  tangential  displacement. 

3rd.  Tangential  force  due  to  radial  displacement. 

4th.  Tangential  force  due  to  tangential  displacement. 

1st.      Put  a'(l+p')  for  a\  and  we  get  the  term  in  p 

— -,  \  ?  ~  ,;  p'  inwards  =  XV>  say (99). 

ira    (a  -af  ^  t^  >      J  v     ^ 


2nd.     By    the    tangential    displacement    of   the    second    ring  the   section   is 
iced   in  the   proportion 
of  the  radial  force  equal  to 


reduced   in  the   proportion   of  1  to   l--j , ,   and  therefore  there  is  an  alteration 


-yr   inwards  =  —  /x'  -j-,  say (100). 


ird'(a  —  a')  ds'  '^   ds' 

3rd.  By  the  radial  displacement  of  the  second  ring  the  direction  of  the 
filament  near  the  part  in  question  is  altered,  so  that  the  attraction  is  no  longer 
radial  but  forwards,  and  the  tangential  part  of  the  force  is 

.5      '^  ^'=+/^'  forwards (lOl). 

ira  (a-a)  ds         '^  ds 

44—2 


348  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

4th.     By  the   tangential   displacement   of  the   second  ring  a  tangential  force 

arises,    depending    on    the   relation   between   the   length   of  the   waves   and  the 

distance  between  the  rings. 

"-ot'  J        f+«xsinp^   , 

If  we   make   m  — -  =  p,  and  m -i  ax  =  H, 

a        ^  J-o.(l+x-y 

/?' 

the  tangential  force  is  a  (a-a'Y  ^^'  ^  *''^'  (102). 

We  may  now  write  down  the  values  of  X,  /x,  and  v  by  transposing  accented 
and  unaccented  letters. 

g^(2a-a)  R  ^^       _?_     n (103). 

ira  (a-aj     '^     TTa{a-a)'        ira  {a-af 

Comparing  these  values  with  those  of  X',  /x',  and  v,  it  will  be  seen  that 
the  following  relations  are  approximately  true  when  a  is  nearly  equal  to  a: 

^'=-'i  =  ^  =  |> (104). 

X  H'      ^      R^ 

27.     To  form  the  equations  of  motion. 

*The  original  equations  were 

^■'  +  o,'p  +  -2o,^-'^,  =  P  =  S+K-(2S-L)Ap-MBp  +  yp--y:'^, 

Putting  p  =  ^  cos  {ills  +  nt),    ar  =  B8m  (ms  +  nt), 

p'  =  A'  cos  {im  +  nt),  cr'^R  sin  {ins  +  nt), 
then  u>'  =  S-vK 

{(o'■V2S+n'-L)A  +  {2(on+M)B-XA'  +  |J:mB  =  0^  ,^^^. 

{2con  +  M)A  +  {n'^-N)B-ij:mA'  +  vR  =  o] ^       '' 

The  corresponding  equations  for  the  second  ring  may  be  found  by  trans- 
posing accented  and  unaccented  letters.  We  should  then  have  four  equations 
to  determine  the  ratios  of  A,  B,  A',  B',  and  a  resultant  equation  of  the  eighth 
degree  to  determine  n.  But  we  may  make  use  of  a  more  convenient  method, 
since   X',  ix,  and  v   are  small.     Eliminating  B  we  find 

An'-A(ai'^-lK+L-N)n'-iAo>Mn  +  AN{Zoy)\_  ,       . 

(-X'A'  +  fx'mR)n'  +  {ix'mA' -v'B')  2<onj        ^       '' 

*  [The  analysis  in  this  article  is  somewhat  unsatisfactory,  the  equations  of  motion  employed  being 
those  which  were  applicable  in  the  case  of  a  ring  of  radius  unity.     Ed.] 


ON    THE    STABILITY    OP    THE    MOTION    OF    SATURN's    RINGS.  349 

Putting  B  =  ^A,  A'  =  xA,  B'  =  ^A'  =  ^xA, 

we  have  ii*  -  {o.'  ( +  2 A")  +  X  - iV}  n' -  4(oMn  +  Sco'N]  ^jj^^       / ^qj^x 

~  =  47i'-2a;';i  +  &c (108), 

an 

-r  =  -  ^''^'  +  H''ml3'}r  +  2/»iw?i  -  2u^'a)n  (109), 

28.  If  we  were  to  solve  the  equation  for  n,  leaving  out  the  terms  involving 
X,  we  should  find  the  wave-velocities  of  the  four  free  waves  of  the  first  ring, 
supposing  the  second  ring  to  be  prevented  from  being  disturbed.  But  in  reality 
the  waves  in  the  first  ring  produce  a  disturbance  in  the  second,  and  these  in 
turn  react  upon  the  first  ring,  so  that  the  wave-velocity  is  somewhat  difierent 
from  that  which  it  would  be  in  the  supposed  case.  Now  if  x  be  the  ratio 
of  the  radial  amplitude  of  displacement  in  the  second  ring  to  that  in  the  first, 
and  if  n  be  a  value  of  n  supposing  cc  =  0,  then  by  Maclaurin's  theorem, 

n=  Jfn  +  -j-x (Ill)- 

The    wave-velocity    relative   to  the   ring  is ,   and    the   absolute   angular 

velocity  of  the  wave  in  space  is 

n  n       I   dn  .   ^-. 

'ar  =  oi =0) j-x (112), 

m  m      m  ax  ' 

=  +p-qx (113), 

,                      n  ,  \   dn 

where  »  =  w ,   and  o  =  —  -j- . 

^  m  ^     m  ax 

Similarly  in  the  second  ring  we  should  have 

-=/-<z'^ (114); 

and   since   the   corresponding  waves   in  the  two  rings   must  have  the  same  abso- 
lute angular  velocity, 

^  =  ■25-',  or  'p  —  qx^'p—ci  - (115)- 


350  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS, 

This  is  a  quadratic  equation  in  x,  the  roots   of  which  are  real  when 

is  positive.  When  this  condition  is  not  fulfilled,  the  roots  are  impossible,  and 
the  general  solution  of  the  equations  of  motion  will  contain  exponential  factors, 
indicating  destructive  oscillations  in  the  rings. 

Since  q  and  q'  are  small  quantities,  the  solution  is  always  real  whenever 
p  and  p'  are  considerably  different.  The  absolute  angular  velocities  of  the  two 
pairs  of  reacting  waves,  are  then  nearly 

V  -\ — ^^/ ,  and  r)  — ^^, , 

instead  of  p  and  p\  as  they  would  have  been  if  there  had  been  no  reaction 
of  the  forced  wave  upon  the  free  wave  which  produces  it. 

When  2^  and  p'  are  equal  or  nearly  equal,  the  character  of  the  solution 
will  depend  on  the  sign  of  qq.  We  must  therefore  determine  the  signs  of  q 
and  q'  in  such  cases. 

Putting  P  =  —7-,  we  may  write  the  values  of  q  and  q' 


x/        ^     /  /6>  fO\         ,,(0     0) 

X  +  211  m    —  -  -   -  4i/  -  - 
n  ^       \n      71/  71  71 


7n '  4?i^  —  2<xr 


Oi         Ct/\  ,       Oi    0) 

,  _  n  ^      \n      71 1 71  n 

^~m"  in"-2o)" 

Referring  to  the  values  of  the  disturbing  forces,  we  find  that 

X'         IX      V  _  Ka 
X  iL      V      Ra" 


(116). 


TT  g      n  471*  — 2&>      Ra  l^^*7\ 

Hence  X  =      _^ —       ,  —-, (117). 

q      n    4n'-2w*     Ra 

Since   qq'  is   of  the   same   sign  as  -^  ,   we   have   only  to  determine  whether 

2  '2 

2n--,   and  2n' -— ,  are  of  the  same  or  of  different  signs.     If  these  quantities 
n  71  ' 

are  of  the  same  sign,  qq   is  positive,  if  of  different  signs,  qq'  is  negative. 


ON    THE    STABILITY     OF    THE    MOTION    OF    SATURN's    RINGS.  351 

Now   there   are    four   values    of  n,    which   give  four   corresponding  values  of 


2n 


72 1=  -W  +  &C.,                  2?ii-      is  negative, 
??j  =  —  a  small  quantity,  2n^ is  positive, 


jjj  =  -f.  a  small  quantity,  211^ is  negative, 

^3 


n^  =  oi  —  kc.,  271^ is  positive. 


The  quantity  with  which  we  have  to  do  is  therefore  positive  for  the  even 
orders  of  waves  and  negative  for  the  odd  ones,  and  the  corresponding  quantity 
in  the  other  ring  obeys  the  same  law.  Hence  when  the  waves  which  act  upon 
each  other  are  either  both  of  even  or  both  of  odd  names,  qq  will  be  positive, 
but  when  one  belongs  to  an  even  series,  and  the  other  to  an  odd  series,  qq 
is  negative. 


29.     The  values  of  j)  and  p'  are,  roughly, 


X>^  =  oi  +  —  —  &c.,  ^o  =  w  +  &c.,  ^3  =  (u  —  &c.,  ^4  =  (o  —  — -  +  &c. 

^j'  =  Co'  H &C.,  p.'  =  0)  +  &c.,  Pa'  =  co'  —  &c.,  Pi=Oi 1-  &C. 


(118). 


<ji  is  greater  than  <u,  so  that  j>^  is  the  greatest,  and  Pi  the  least  of  these 
values,  and  of  those  of  the  same  order,  the  accented  is  greater  than  the  unac- 
cented. The  following  cases  of  equahty  are  therefore  possible  under  suitable 
circumstances ; 

P,  =P,\  Pi  =p/» 

P4=P,'  (when  m=l),   p,=2^3, 

p.=p:, 

In  the  cases  in  the  first  column  qq'  will  be  positive,  in  those  in  the  second 
column  qq'  will  be  negative. 


352  ox    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

30.  Now  each  of  the  four  values  of  p  is  a  function  of  w,  the  number 
of  undulations  in  the  ring,  and  of  a  the  radius  of  the  ring,  varying  nearly 
as  cfl  Hence  m  being  given,  we  may  alter  the  radius  of  the  ring  till  any 
one  of  the  four  values  of  p  becomes  equal  to  a  given  quantity,  say  a  given 
value  of  /,  so  that  if  an  indefinite  number  of  rings  coexisted,  so  as  to  form 
a  sheet  of  rings,  it  would  be  always  possible  to  discover  instances  of  the 
equality  of  x>  ^^^  V  among  them.  K  such  a  case  of  equahty  belongs  to  the 
first  column  given  above,  two  constant  waves  will  arise  in  both  rings,  one 
travelling  a  little  faster,  and  the  other  a  little  slower  than  the  free  waves. 
If  the  case  belongs  to  the  second  column,  two  waves  will  also  arise  in  each 
ring,  but  the  one  pair  will  graduaUy  die  away,  and  the  other  pair  wHl  increase 
in  ampUtude  indefinitely,  the  one  wave  strengthening  the  other  till  at  last  both 
rino-s  are  thrown  into  confusion. 

The  only  way  in  which  such  an  occurrence  can  be  avoided  is  by  placing 
the  rings  at  such  a  distance  that  no  value  of  m  shall  give  coincident  values 
of  _p  and  J),  For  instance,  if  w  >  2a),  but  w  <  So),  no  such  coincidence  is  possible. 
For  j)^  is  always  less  than  p./,  it  is  greater  than  p,  when  m  =  1  or  2,  and  less 
than  _p4  when  m  is  3  or  a  greater  number.  There  are  of  course  an  infinite 
number  of  ways  in  which  this  noncoincidence  might  be  secured,  but  it  is  plain 
that  if  a  number  of  concentric  rings  were  placed  at  small  intervals  from  each 
other,  such  coincidences  must  occur  accurately  or  approximately  between  some 
pairs' of  rings,  and  if  the  value  of  [p-fj  is  brought  lower  than  -^qq,  there 
will  be  destructive  interference. 

This  investigation  is  applicable  to  any  number  of  concentric  rings,  for,  by 
the  principle  of  superposition  of  small  displacements,  the  reciprocal  actions  of 
any  pair  of  rings  are  independent  of  all  the  rest. 

31.     On  the  effect  of  long-continued  disturbances  on  a  system  of  rings. 

The  result  of  our  previous  investigations  has  been  to  point  out  several 
ways  in  which  disturbances  may  accumulate  till  collisions  of  the  different  par- 
ticles of  the  rings  take  place.  After  such  a  collision  the  particles  wUl  still 
continue  to  revolve  about  the  planet,  but  there  will  be  a  loss  of  energy  in 
the  system  during  the  colUsion  which  can  never  be  restored.  Such  coUisions 
however  will  not  affect  what  is  called  the  Angular  Momentum  of  the  system 
about  the  planet,  which  will  therefore  remain  constant. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  353 

Let  M  be  the  mass  of  tlie  system  of  rings,  and  hm  that  of  one  ring 
whose  radius  is  r,  and  angular  velocity  (o  =  S^r~^.  The  angular  momentum  of 
the  ring  is 

half  its  vis  viva  is  ^tuV'Sm  =  ^Sr~^  hm. 

The  potential  energy  due  to  Saturn's  attraction  on  the  ring  is 

-Sr-'hm. 
The  angular  momentum  of  the  whole  system  is  invariable,  and  is 

S'^%{r^hm)  =  A (119). 

The  whole  energy  of  the  system  is  the  sum  of  half  the  vis  viva  and  the 
potential  energy,  and  is 

-^St{r-'hm)  =  E (120). 

A  is  invariable,  while  E  necessarily  diminishes.  We  shall  find  that  as  E 
diminishes,  the  distribution  of  the  rings  must  be  altered,  some  of  the  outer 
rings  moving  outwards,  while  the  inner  rings  move  inwards,  so  as  either  to 
spread  out  the  whole  system  more,  both  on  the  outer  and  on  the  inner  edge 
of  the  system,  or,  without  affecting  the  extreme  rings,  to  diminish  the  density 
or  number  of  the  rings  at  the  mean  distance,  and  increase  it  at  or  near  the 
inner  and  outer  edges. 


Let  us  put  x  =  r^-, 

then  A- 

=  S-t{xdm)  is  constant. 

Now  let 

tixdm) 
^^~  t{dm)  ' 

and 

X  =  Xi  +  x\ 

then  we  may  write 

-^  =  t(r-^Bm)=^t{x-'dm), 
lb 


=  Sc^m(a:--2|3  +  3|i-&c.), 


=  \t{dm)-^,X{xdm)-]-^,t(x'Bm)-kc (121). 


354  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

Now  t(dm)  =  M  a  constant,  t(xdm)  =  0,  and  t(x"-Bm)  is  a  quantity  which 
increases  when  the  rings  are  spread  out  from  the  mean  distance  either  way, 
X  being  subject  only  to  the  restriction  t  (xdm)  =  0.  But  %  (x'dm)  may 
increase  without  the  extreme  values  of  x  being  increased,  provided  some  other 
values  be  increased. 

32.  In  fact,  if  we  consider  the  very  innermost  particle  as  moving  in  an 
ellipse,  and  at  the  further  apse  of  its  orbit  encountering  another  particle 
belonging  to  a  larger  orbit,  we  know  that  the  second  particle,  when  at  the 
same  distance  from  the  planet,  moves  the  faster.  The  result  is,  that  the 
interior  satellite  will  receive  a  forward  impulse  at  its  further  apse,  and  will 
move  in  a  larger  and  less  eccentric  orbit  than  before.  In  the  same  way  one 
of  the  outermost  particles  may  receive  a  backward  impulse  at  its  nearer  apse, 
and  so  be  made  to  move  in  a  smaller  and  less  eccentric  orbit  than  before. 
When  we  come  to  deal  with  collisions  among  bodies  of  unknown  number,  size, 
and  shape,  we  can  no  longer  trace  the  mathematical  laws  of  their  motion  with 
any  distinctness.  All  we  can  now  do  is  to  collect  the  results  of  our  investi- 
gations and  to  make  the  best  use  we  can  of  them  in  forming  an  opinion  as 
to  the  constitution  of  the  actual  rings  of  Saturn  which  are  still  in  existence 
and  apparently  in  steady  motion,  whatever  catastrophes  may  be  indicated  by 
the  various  theories  we  have  attempted. 

33.  To  find  the  Loss  of  Energy  due  to  internal  friction  in  a  hroad  Fluid 
Ring,  the  parts  of  which  revolve  about  the  Planet,  each  with  the  velocity  of  a 
satellite  at  the  same  distance. 

Conceive  a  fluid,  the  particles  of  which  move  parallel  to  the  axis  of  x 
with  a  velocity  u,  u  being  a  function  of  z,  then  there  will  be  a  tangential  pres- 
sure on  a  plane  parallel  to  xy 

dU  ..  r. 

=  /x-y-  on  umt  01  area 
'^  dz 

due  to  the  relative  sliding  of  the  parts  of  the  fluid  over  each  other. 
In  the  case  of  the  ring  we  have 

The  absolute  velocity  of  any  particle  is  tor.  That  of  a  particle  at  distance 
{r-\-Zr)  is 

(ar  +  -j-  {(ar)  hr. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's     RINGS.  355 

If  the  angular  velocity  had  been  uniform,  there  would  have  been  no  sliding, 
and  the  velocity  would  have  been 

cji"  +  (ohr. 
The  sliding  is  therefore 

d(o  ^ 

r  -J-  or, 
ar 

and  the  friction  on  unit  of  area  perpendicular  to  r  is  fir  -p  • 

The  loss  of  Energy,  per   unit   of  area,   is   the   product  of  the  sliding  by  the 
friction, 

or,  /x?-*-^    Sr  in  unit  of  time. 

The   loss    of    Energy  in   a   part    of    the    Ring    whose   radius    is   r,    breadth 
Sr,  and  thickness  c,  is 

27rr*c/x  -j-    Sr. 
In  the  case  before  us  it  is  f  Tr/x/Scr"*  Sr. 

If  the   thickness  of  the  ring   is  uniform   between  r  =  a  and   r  =  h,  the    whole 
loss  of  Energy  is 

in  unit  of  time. 

Now  half  the  vis  viva  of  an  elementary  ring  is 
npcrhr  r^oy  =  nfxSSr, 
and  this  between  the  limits  r  =  a  and  r  =  h  gives 

npcS  (a  —  h). 

The  potential   due   to   the  attraction   of  5  is   twice   this   quantity    with   the 
sign  changed,  so  that 

E=-TrpcS(a-b), 

E  dt~      ^  p  ah' 

45—2 


356  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

Now  Professor  Stokes  finds  a/^  =  0-0564  for  water, 

^  P 

and  =0'116  for  air, 

taking  the  unit  of  space  one  English  inch,  and  the  unit  of  time  one  second. 
We  may  take  a  =  88,209  miles,  and  ?>  =  77,636  for  the  ring  A)  and  a  =  75,845, 
and  6  =  58,660  for  the  ring  B.  We  may  also  take  one  year  as  the  unit  of 
time.  The  quantity  representing  the  ratio  of  the  loss  of  energy  in  a  year  to 
the  whole  energy  is 

I   dE  1  p      .-L       •        ^ 

E  W=  60,880,000,000,000  ^^'  ^^'  "^^  ^' 

^^  39,540,000,000,000  ^''  ^^^  ^^"^  ^' 

showing  that  the  efiect  of  internal  friction  in  a  ring  of  water  moving  with 
steady  motion  is  inappreciably  small.  It  cannot  be  from  this  cause  therefore 
that  any  decay  can  take  place  in  the  motion  of  the  ring,  provided  that  no 
waves  arise  to  disturb  the  motion. 


Recapitulation  of  the  Tlieory  of  the  Motion  of  a  Rigid  Ring. 

The  position  of  the  ring  relative  to  Saturn  at  any  given  instant  is  defined 
by  three  variable  quantities. 

1st.  The  distance  between  the  centre  of  gravity  of  Saturn  and  the  centre 
of  gravity  of  the  ring.     This  distance  we  denote   by  r. 

2nd.  The  angle  which  the  line  r  makes  with  a  fixed  line  in  the  plane  of 
the  motion  of  the  ring.    This  angle  is  called  0. 

3rd.  The  angle  between  the  line  r  and  a  Hne  fixed  with  respect  to  the 
ring  so  that  it  coincides  with  r  when  the  ring  is  in  its  mean  position.  This  is 
the  angle  <^. 

The  values  of  these  three  quantities  determine  the  position  of  the  ring  so 
far  as  its  motion  in  its  own  plane  is  concerned.  They  may  be  referred  to  as 
the  radius  vector,   longitude,  and  angle  of  lihration  of  the  ring. 

The  forces  which  act  between  the  ring  and  the  planet  depend  entirely  upon 
their  relative  positions.     The  method   adopted   above   consists  in  determining  the 


ON    THE    STABILITY    OF    THE    MOTION     OF    SATURN's     RINGS.  357 

potential  ( V)  of  the  ring  at  the  centre  of  the  planet  In  terms  of  r  and  <^.     Then 

the  work  done  by   any  displacement   of  the   system    is   measured   by   the   change 

of   VS  during  that  displacement.     The   attraction   between   the   centre   of  gravity 

(IV 
of  the  Ring  and  that  of  the  planet  Is    ~S   ,   ,   and   the   moment  of  the  couple 

clV 
tending  to   turn   the  ring  about  Its  centre   of  gravity  Is  S-j-j, 

It  Is  proved  In  Problem  V,  that  if  a  be  the  radius  of  a  circular  ring,  r^^uf 
the   distance   of  its   centre   of  gravity  from   the  centre   of  the  circle,  and  R  the 

mass  of  the  ring,  then,  at  the  centre  of  the  ring,     ,-  = 5/,  -yj  =  0. 

(PV       Ji 
It  also  appears  that    T-^  =  -k~3  {^  +9)>  "which  is  positive  when  g  >  —I, 

d'V       R 
and  that   -n\=^—f'(^—g),   which  is  positive  when  ^<3. 

d'V  .  .  . 

If  -y—  is  positive,  then  the  attraction  between  the  centres  decreases  as  the 

distance   increases,    so  that,   if  the   two    centres    were    kept    at    rest   at   a   given 

d'V  .         .  . 
distance  by  a  constant  force,  the  equilibrium  would  be  unstable.     If  -t-t;  is  positive, 

then  the  forces  tend  to  increase  the  angle  of  libration,  in  whichever  direction 
the  libration  takes  place,  so  that  if  the  ring  were  fixed  by  an  axis  through  its 
centre  of  gravity,  its  equilibrium  round  that  axis  would  be  unstable. 

In  the  case  of  the  uniform  ring  with  a  heavy  particle  on  its  circumference 
whose  weight  ="82  of  the  whole,  the  direction  of  the  whole  attractive  force  of 
the  ring  near  the  centre  will  pass  through  a  point  lying  in  the  same  radius  as 
the  centre  of  gravity,   but  at  a  distance  from  the  centre  =  fa.     (Fig.  6.) 

If  we  call  this  point  0,  the  line  SO  will  indicate  the  direction  and  position 
of  the  force  acting  on  the  ring,  which   we  may  call  F. 

It  Is  evident  that  the  force  F,  acting  on  the  ring  in  the  line  OS,  will  tend 
to  turn  it  round  its  centre  of  gravity  R  and  to  increase  the  angle  of  libration 
KRO.  The  direct  action  of  this  force  can  never  reduce  the  angle  of  libration 
to  zero  again.  To  understand  the  indirect  action  of  the  force,  we  must  recollect 
that  the  centre  of  gravity  (i?)  of  the  ring  is  revolving  about  Saturn  in  the 
direction  of  the  arrows,  and  that  the  ring  is  revolving  about  its  centre  of  gravity 


358  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

with  nearly  the  same  velocity.  If  the  angular  velocity  of  the  centre  of  gravity 
about  Saturn  were  always  equal  to  the  rotatory  velocity  of  the  ring,  there 
would  be  no  libration. 

Now  suppose  that  the  angle  of  rotation  of  the  ring  is  in  advance  of  the 
longitude  of  its  centre  of  gravity,  so  that  the  line  RO  has  got  in  advance  of 
SRK  by  the  angle  of  libration  KRO.  The  attraction  between  the  planet  and 
the  ring  is  a  force  F  acting  in  SO.  We  resolve  this  force  into  a  couple,  whose 
moment  is  FRN,  and  a  force  F  acting  through  R  the  centre  of  gravity  of  the 
ring. 

The  couple  affects  the  rotation  of  the  ring,  but  not  the  position  of  its  centre 
of  gravity,  and  the  force  RF  acts  on  the  centre  of  gravity  without  affecting  the 
rotation. 

Now  the  couple,  in  the  case  represented  in  the  figure,  acts  in  the  positive 
direction,  so  as  to  increase  the  angular  velocity  of  the  ring,  which  was  already 
greater  than  the  velocity  of  revolution  of  R  about  S,  so  that  the  angle  of 
libration  would  increase,  and  never  be  reduced  to  zero. 

The  force  RF  does  not  act  in  the  direction  of  >S',  but  behind  it,  so  that  it 
becomes  a  retarding  force  acting  upon  the  centre  of  gravity  of  the  ring.  Now 
the  effect  of  a  retarding  force  is  to  cause  the  distance  of  the  revolving  body  to 
decrease  and  the  angular  velocity  to  increase,  so  that  a  retarding  force  increases 
the  angular  velocity  of  R  about  S. 

The  effect  of  the  attraction  along  SO  in  the  case  of  the  figure  is,  first,  to 
increase  the  rate  of  rotation  of  the  ring  round  R,  and  secondly,  to  iacrease  the 
angular  velocity  of  R  about  S.  If  the  second  effect  is  greater  than  the  first, 
then,  although  the  line  RO  increases  its  angular  velocity,  SR  will  increase  its 
angular  velocity  more,  and  will  overtake  RO,  and  restore  the  ring  to  its  original 
position,  so  that  SRO  will  be  made  a  straight  line  as  at  first.  If  this  accelerat- 
ing effect  is  not  greater  than  the  acceleration  of  rotation  about  R  due  to  the 
couple,  then  no  compensation  will  take  place,  and  the  motion  will  be  essentially 
unstable. 

If  in  the  figure  we  had  drawn  ^  negative  instead  of  positive,  then  the 
couple  would  have  been  negative,  the  tangential  force  on  R  accelerative,  r  would 
have  increased,  and  in  the  cases  of  stability  the  retardation  of  6  would  be  greater 
than  that  of  (^  +  <^),  and  the  normal  position  would  be  restored,  as  before. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  359 

The  object  of  the  investigation  is  to  find  the  conditions  under  wliich  this 
compensation  is  possible. 

It  is  evident  that  when  SRO  becomes  straight,  there  is  still  a  difference 
of  angular  velocities  between  the  rotation  of  the  ring  and  the  revolution  of 
the  centre  of  gravity,  so  that  there  will  be  an  oscillation  on  the  other  side, 
and  the  motion  will  proceed  by  alternate  oscillations  without  limit. 

If  we  begin  with  r  at  its  mean  value,  and  <^  negative,  then  the  rotation 
of  the  ring  will  be  retarded,  7*  will  be  increased,  the  revolution  of  r  will  be 
more  retarded,  and  thus  <f>  will  be  reduced  to  zero.  The  next  part  of  the 
motion  will  reduce  r  to  its  mean  value,  and  bring  (f)  to  its  greatest  positive 
value.  Then  r  will  diminish  to  its  least  value,  and  (f>  will  vanish.  Lastly  r 
will  return  to  the  mean  value,  and  <f)  to  the  greatest  negative  value. 

It  appears  from  the  calculations,  that  there  are,  in  general,  two  different 
ways  in  which  this  kind  of  motion  may  take  place,  and  that  these  may  have 
different  periods,  phases,  and  amplitudes.  The  mental  exertion  required  in  follow- 
ing out  the  results  of  a  combined  motion  of  this  kind,  with  all  the  variations  of 
force  and  velocity  during  a  complete  cycle,  w^ould  be  very  great  in  proportion  to 
the  additional  knowledge  we  should  derive  from  the  exercise. 

The  result  of  this  theory  of  a  rigid  ring  shows  not  only  that  a  perfectly 
uniform  ring  cannot  revolve  permanently  about  the  planet,  but  that  the  irregu- 
larity of  a  permanently  revolving  ring  must  be  a  very  observable  quantity,  the 
distance  between  the  centre  of  the  ring  and  the  centre  of  gravity  being  between 
•8158  and  '8279  of  the  radius.  As  there  is  no  appearance  about  the  rings 
justifying  a  belief  in  so  great  an  irregularity,  the  theory  of  the  solidity  of  the 
rings  becomes  very  improbable. 

When  we  come  to  consider  the  additional  difficulty  of  the  tendency  of  the 
fluid  or  loose  parts  of  the  ring  to  accumulate  at  the  thicker  parts,  and  thus 
to  destroy  that  nice  adjustment  of  the  load  on  which  stability  depends,  we 
have  another  powerful  argument  against  solidity. 

And  when  we  consider  the  immense  size  of  the  rings,  and  their  comparative 
thinness,  the  absurdity  of  treating  them  as  rigid  bodies  becomes  self-evident. 
An  iron  ring  of  such  a  size  would  be  not  only  plastic  but  semifluid  under  the 
forces  which  it  would  experience,  and  we  have  no  reason  to  believe  these  rings 
to  be  artificially  strengthened  with  any  material  unknown  on  this  earth. 


360  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 


Recapitulation  of  the  Theory  of  a  Ring  of  equal  Satellites. 

In  attempting  to  conceive  of  the  disturbed  motion  of  a  ring  of  unconnected 
satellites,  we  have,  in  the  first  place,  to  devise  a  method  of  identifying  each 
satellite  at  any  given  time,  and  in  the  second  place,  to  express  the  motion  of 
every  satellite  under  the  same  general  formula,  in  order  that  the  mathematical 
methods  may  embrace  the  whole  system  of  bodies  at  once. 

By  conceiving  the  ring  of  satellites  arranged  regularly  in  a  circle,  we  may 
easily  identify  any  satellite,  by  stating  the  angular  distance  between  it  and  a 
known  satellite  when  so  arranged.  If  the  motion  of  the  ring  were  undisturbed, 
this  angle  would  remain  unchanged  during  the  motion,  but,  in  reality,  the 
satellite  has  its  position  altered  in  three  ways :  1st,  it  may  be  further  from 
or  nearer  to  Saturn;  2ndly,  it  may  be  in  advance  or  in  the  rear  of  the  position 
it  would  have  had  if  undisturbed ;  3rdly,  it  may  be  on  one  side  or  other  of 
the  mean  plane  of  the  ring.  Each  of  these  displacements  may  vary  in  any  way 
whatever  as  we  pass  from  one  satellite  to  another,  so  that  it  is  impossible 
to  assign  beforehand  the  place  of  any  satellite  by  knowing  the  places  of  the 
rest.     §  2. 

The  formula,  therefore,  by  which  we  are  enabled  to  predict  the  place  of 
every  satellite  at  any  given  time,  must  be  such  as  to  allow  the  initial  position 
of  every  satellite  to  be  independent  of  the  rest,  and  must  express  all  future 
positions  of  that  satellite  by  inserting  the  corresponding  value  of  the  quantity 
denoting  time,  and  those  of  every  other  sateUite  by  inserting  the  value  of  the 
angular  distance  of  the  given  satelUte  from  the  point  of  reference.  The  three 
displacements  of  the  satellite  will  therefore  be  functions  of  two  variables — the 
angular  position  of  the  satellite,  and  the  time.  When  the  time  alone  is  made 
to  vary,  we  trace  the  complete  motion  of  a  single  satellite ;  and  when  the  time 
is  made  constant,  and  the  angle  is  made  to  vary,  we  trace  the  form  of  the 
ring  at  a  given  time. 

It  is  evident  that  the  fonn  of  this  function,  in  so  far  as  it  indicates  the 
state  of  the  whole  ring  at  a  given  instant,  must  be  wholly  arbitrary,  for  the 
form  of  the  ring  and  its  motion  at  starting  are  limited  only  by  the  condition 
that  the  irregularities  must  be  small.  We  have,  however,  the  means  of  breaking 
up  any  function,  however  complicated,  into  a  series  of  simple  functions,  so  that 
the   value   of    the   function   between    certain    limits   may   be   accurately  expressed 


ON    THE    STABILITY     OF    THE    MOTION     OF    SATURN's    RINGS.  361 

as  the  sum  of  a  series  of  sines  and  cosines  of  multiples  of  the  variable.  This 
method,  due  to  Fourier,  is  peculiarly  applicable  to  the  case  of  a  ring  returning 
into  itself,  for  the  value  of  Fourier's  series  is  necessarily  periodic.  We  now 
regard  the  form  of  the  disturbed  ring  at  any  instant  as  the  result  of  the 
superposition  of  a  number  of  separate  disturbances,  each  of  -which  is  of  the  nature 
of  a  series  of  equal  waves  regularly  arranged  round  the.  ring.  Each  of  these 
elementary  disturbances  is  characterised  by  the  number  of  undulations  in  it,  by 
their  amplitude,  and  by  the  position  of  the  first  maximum  in  the  ring.     §  3. 

When  we  know  the  form  of  each  elementary  disturbance,  we  may  calculate 
the  attraction  of  the  disturbed  ring  on  any  given  particle  in  terms  of  the  con- 
stants belonging  to  that  disturbance,  so  that  as  the  actual  displacement  is  the 
resultant  of  the  elementary  displacements,  the  actual  attraction  will  be  the 
resultant  of  the  corresponding  elementary  attractions,  and  therefore  the  actual 
motion  will  be  the  resultant  of  all  the  motions  arising  from  the  elementary 
disturbances.  We  have  therefore  only  to  investigate  the  elementary  disturbances 
one  by  one,  and  having  established  the  theory  of  these,  we  calculate  the  actual 
motion  by  combining  the  series  of  motions  so  obtained. 

Assuming  the  motion  of  the  satellites  in  one  of  the  elementary  disturbances 
to  be  that  of  oscillation  about  a  mean  position,  and  the  whole  motion  to  be 
that  of  a  uniformly  revolving  series  of  undulations,  we  find  our  supposition  to 
be  correct,  provided  a  certain  biquadratic  equation  is  satisfied  by  the  quantity 
denoting  the  rate  of  oscillation.     §  6. 

When  the  four  roots  of  this  equation  are  all  real,  the  motion  of  each 
satellite  is  compounded  of  four  difierent  oscillations  of  difi'erent  amplitudes  and 
periods,  and  the  motion  of  the  whole  ring  consists  of  four  series  of  undulations, 
travelling  round  the  ring  with  different  velocities.  When  any  of  these  roots 
are  impossible,  the  motion  is  no  longer  oscillatory,  but  tends  to  the  rapid 
destruction  of  the  ring. 

To  determine  whether  the  motion  of  the  ring  is  permanent,  we  must  assure 
ourselves  that  the  four  roots  of  this  equation  are  real,  whatever  be  the  number 
of  undulations  in  the  ring;  for  if  any  one  of  the  possible  elementary  distui'b- 
ances  should  lead  to  destructive  oscillations,  that  disturbance  might  sooner  or 
later  commence,  and  the  ring  would  be  destroyed. 

Now  the  number  of  undulations  in  the  ring  may  be  any  whole  number 
from   one    up   to  half  the   number   of  satellites.      The   forces   from   which   danger 


362  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

is  to  be  apprehended  are  greatest  when  the  number  of  undulations  is  greatest, 
and  by  taking  that  number  equal  to  half  the  number  of  satellites,  we  find  the 
condition  of  stability  to  be 

S>.A352tiR, 

where  S  is  the  mass  of  the  central  body,  R  that  of  the  ring,  and  /x  the  number 
of  sateUites  of  which  it  is  composed.  §  8.  If  the  number  of  satelHtes  be  too 
great,  destructive  oscillations  will  commence,  and  finally  some  of  the  satellites 
will  come  into  coUision  with  each  other  and  unite,  so  that  the  number  of 
independent  satellites  will  be  reduced  to  that  which  the  central  body  can  retain 
and  keep  in  discipline.  When  this  has  taken  place,  the  satellites  will  not  only 
be  kept  at  the  proper  distance  from  the  primary,  but  will  be  prevented  by  its 
preponderating  mass  from  interfering  with  each  other. 

We  next  considered  more  carefully  the  case  in  which  the  mass  of  the  ring 
is  very  small,  so  that  the  forces  arising  from  the  attraction  of  the  ring  are 
small  compared  with  that  due  to  the  central  body.  In  this  case  the  values 
of  the  roots  of  the  biquadratic  are  all  real,  and  easUy  estimated.     §  9. 

If  we  consider  the  motion  of  any  satellite  about  its  mean  position,  as 
referred  to  axes  fixed  in  the  plane  of  the  ring,  we  shall  find  that  it  describes 
an  ellipse  in  the  direction  opposite  to  that  of  the  revolution  of  the  ring,  the 
periodic  time  being  to  that  of  the  ring  as  o>  to  n,  and  the  tangential  ampli- 
tude of  oscillation  being  to  the  radial  as  2(0  to  n.     §  10. 

The  absolute  motion  of  each  satellite  in  space  is  nearly  elliptic  for  the  large 
values  of  n,  the  axis  of  the  ellipse  always  advancing  slowly  in  the  direction  of 
rotation.  The  path  of  a  satellite  corresponding  to  one  of  the  small  values  of 
n  is  nearly  circular,  but  the  radius  slowly  increases  and  diminishes  during  a 
period  of  many  revolutions.     §  11. 

The  form  of  the  ring  at  any  instant  is  that  of  a  re-entering  curve,  having 
m  alternations  of  distance  from  the  centre,  symmetrically  arranged,  and  m  points 
of  condensation,  or  crowding  of  the  satellites,  which  coincide  with  the  points  of 
greatest  distance  when  n  is  positive,  and  with  the  points  nearest  the  centre 
when  n  m  negative.     §  12. 

This  system  of  undulations  travels  with  an  angular  velocity relative   to 

the   ring,   and   co in   space,    so   that   during   each   oscillation    of   a   satellite   a 

complete  wave  passes  over  it.     §  14. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  363 

To  exhibit  the  movements  of  the  satellites,  I  have  made  an  arrangement 
by  which  36  little  ivory  balls  are  made  to  go  through  the  motions  belonging 
to  the  first  or  fourth  series  of  waves.     (Figs.  7,  8.) 

The  instrument  stands  on  a  pillar  A,  in  the  upper  part  of  which  turns 
the  cranked  axle  CC.  On  the  parallel  parts  of  this  axle  are  placed  two  wheels, 
RR  and  TT,  each  of  which  has  36  holes  at  equal  distances  in  a  circle  neai- 
its  circumference.  The  two  circles  are  connected  by  36  small  cranks  of  the 
fonn  KK,  the  extremities  of  which  turn  in  the  corresponding  holes  of  the  two 
wheels.  That  axle  of  the  crank  K  which  passes  through  the  hole  in  the  wheel 
S  is  bored,  so  as  to  hold  the  end  of  the  bent  wire  which  carries  the  satellite  >S'. 
This  wire  may  be  turned  in  the  hole  so  as  to  place  the  bent  part  carrying 
the  satellite  at  any  angle  with  the  crank.  A  pin  F,  which  passes  through  the 
top  of  the  pillar,  serves  to  prevent  the  cranked  axle  from  turning ;  and  a  pin  Q, 
passing  through  the  pillar  horizontally,  may  be  made  to  fix  the  wheel  R,  by 
inserting  it  in  a  hole  in  one  of  the  spokes  of  that  wheel.  There  is  also  a 
handle  H,  which  is  in  one  piece  with  the  wheel  T,  and  serves  to  turn  the  axle. 

Now  suppose  the  pin  P  taken  out,  so  as  to  allow  the  cranked  axle  to 
turn,  and  the  pin  Q  inserted  in  its  hole,  so  as  to  prevent  the  wheel  R  from 
revolving;  then  if  the  crank  C  be  turned  by  means  of  the  handle  H,  the 
wheel  T  will  have  its  centre  carried  round  in  a  vertical  circle,  but  will  remain 
parallel  to  itself  during  the  whole  motion,  so  that  every  point  in  its  plane  will 
describe  an  equal  circle,  and  all  the  cranks  K  will  be  made  to  revolve  exactly 
as  the  large  crank  C  does.  Each  satellite  will  therefore  revolve  in  a  small 
circular  orbit,  in  the  same  time  with  the  handle  H,  but  the  position  of  each 
satellite  in  that  orbit  may  be  arranged  as  we  please,  according  as  we  turn  the 
wire  which  supports  it  in  the  end  of  the  crank. 

In  fig.  8,  which  gives  a  front  view  of  the  instrument,  the  satelHtes  are  so 
placed  that  each  is  turned  60^  further  round  in  its  socket  than  the  one  behind 
it.  As  there  are  36  satellites,  this  process  will  bring  us  back  to  our  starting- 
point  after  six  revolutions  of  the  direction  of  the  arm  of  the  satellite;  and 
therefore  as  we  have  gone  round  the  ring  once  in  the  same  direction,  the  ami 
of  the   sateUite  will  have  overtaken  the  radius  of  the  ring  five  times. 

Hence  there  will  be  five  places  where  the  satellites  are  beyond  their  mean 
distance  from  the  centre  of  the  ring,  and  five  where  they  are  within  it,  so 
that   we   have   here   a   series  of  five    undulations   round  the  circumference  of  the 

46—2 


364  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

ring.     In  this  case  the  satellites  are  crowded  together  when  nearest  to  the  centre, 
so  that  the  case  is  that  of  the  first  series  of  waves,   when  m  =  5. 

Now  suppose  the  cranked  axle  C  to  be  turned,  and  all  the  small  cranks 
K  to  turn  with  it,  as  before  explained,  every  satellite  will  then  be  carried 
round  on  its  own  arm  in  the  same  direction ;  but,  since  the  direction  of  the 
arms  of  different  satellites  is  different,  their  phases  of  revolution  will  preserve 
the  same  difference,  and  the  system  of  satellites  will  still  be  arranged  in  five 
undulations,  only  the  undulations  will  be  propagated  round  the  ring  in  the 
direction   opposite   to   that   of  the   revolution   of  the   satellites. 

To  understand  the  motion  better,  let  us  conceive  the  centres  of  the  orbits 
of  the  satellites  to  be  arranged  in  a  straight  line  instead  of  a  circle,  as  in 
fig.  10.  Each  satellite  is  here  represented  in  a  different  phase  of  its  orbit,  so 
that  as  we  pass  from  one  to  another  from  left  to  right,  we  find  the  position 
of  the  satellite  in  its  orbit  altering  in  the  direction  opposite  to  that  of  the 
hands  of  a  watch.  The  satellites  all  lie  in  a  trochoidal  curve,  indicated  by 
the  line  through  them  in  the  figure.  Now  conceive  every  satellite  to  move  in 
its  orbit  through  a  certain  angle  in  the  direction  of  the  arrows.  The  satellites 
will  then  lie  in  the  dotted  line,  the  form  of  which  is  the  same  as  that  of 
the  former  curve,  only  shifted  in  the  direction  of  the  large  arrow.  It  appears, 
therefore,  that  as  the  satellites  revolve,  the  undulation  travels,  so  that  any 
part  of  it  reaches  successively  each  satellite  as  it  comes  into  the  same  phase 
of  rotation.  It  therefore  travels  from  those  satellites  which  are  most  advanced 
in  phase  to  those  which  are  less  so,  and  passes  over  a  complete  wave-length 
in  the   time   of  one  revolution   of  a   satellite. 

Now  if  the  satellites  be  arranged  as  in  fig.  8,  where  each  is  more  advanced 
in  phase  as  we  go  round  the  ring  in  the  direction  of  rotation,  the  wave  will 
travel  in  the  direction  opposite  to  that  of  rotation,  but  if  they  are  arranged 
as  in  fig.  12,  where  each  satellite  is  less  advanced  in  phase  as  we  go  round 
the  ring,  the  wave  will  travel  in  the  direction  of  rotation.  Fig.  8  represents 
the  first  series  of  waves  where  m  =  5,  and  fig.  12  represents  the  fourth  series 
where  m  =  7.  By  arranging  the  satellites  in  their  sockets  before  starting,  we 
might  make  w  equal  to  any  whole  number,  from  1  to  18.  If  we  chose  any 
number  above  18  the  result  would  be  the  same  as  if  we  had  taken  a  number 
as  much  below  18  and  changed  the  arrangement  from  the  first  wave  to  the 
fourth. 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S     RINGS.  365 

In  this  way  we  can  exhibit  the  motions  of  the  satellites  in  the  first  and 
fourth  waves.  In  reality  they  ought  to  move  in  ellipses,  the  major  axes  being 
twice  the  minor,  whereas  in  the  machine  they  move  in  circles :  but  the  character 
of  the  motion  is  the  same,  though  the  form  of  the  orbit  is  diflferent. 

We  may  now  show  these  motions  of  the  satellites  among  each  other,  com- 
bined with  the  motion  of  rotation  of  the  whole  ring.  For  this  purpose  we 
put  in  the  pin  P,  so  as  to  prevent  the  crank  axle  from  turning,  and  take 
out  the  pin  ^  so  as  to  allow  the  wheel  R  to  turn.  If  we  then  turn  the 
wheel  T,  all  the  small  cranks  will  remain  parallel  to  the  fixed  crank,  and  the 
wheel  R  will  revolve  at  the  same  rate  as  T.  The  arm  of  each  satellite  will 
continue  parallel  to  itself  during  the  motion,  so  that  the  satellite  will  describe 
a  circle  whose  centre  is  at  a  distance  from  the  centre  of  R,  equal  to  the  arm 
of  the  satellite,  and  measured  in  the  same  direction.  In  our  theory  of  real 
satellites,  each  moves  in  an  ellipse,  having  the  central  body  in  its  focus,  but 
this  motion  in  an  eccentric  circle  is  sufficiently  near  for  illustration.  The 
motion  of  the  waves  relative  to  the  ring  is  the  same  as  before.  The  waves 
of  the  first  kind  travel  faster  than  the  ring  itself,  and  overtake  the  satellites, 
those   of  the  fourth  kind  travel  slower,  and  are  overtaken  by  them. 

In  fig.  11  we  have  an  exaggerated  representation  of  a  ring  of  twelve  satel- 
lites afiected  by  a  wave  of  the  fourth  kind  where  m  =  2.  The  satellites  here  lie  in 
an  eUipse  at  any  given  instant,  and  as  each  moves  round  in  its  circle  about 
its  mean  position,  the  ellipse  also  moves  round  in  the  same  direction  with  half 
their  angular  velocity.  In  the  figure  the  dotted  line  represents  the  position  of 
the  ellipse  when  each  satellite  has  moved  forward  into  the  position  represented 
by  a  dot. 

Fig.  13  represents  a  wave  of  the  first  kind  where  m  =  2.  The  satellites  at 
any  instant  lie  in  an  epitrochoid,  which,  as  the  satellites  revolve  about  their 
mean  positions,  revolves  in  the  opposite  direction  with  half  their  angular  velocity, 
so  that  when  the  satellites  come  into  the  positions  represented  by  the  dots, 
the  curve  in  which  they  lie  turns  round  in  the  opposite  direction  and  forms  the 
dotted  curve. 

In  fig.  9  we  have  the  same  case  as  in  fig.  13,  only  that  the  absolute  orbits 
of  the  satellites  in  space  are  given,  instead  of  their  orbits  about  their  mean 
positions  in  the  ring.     Here  each  moves  about  the  central    body   in  an   eccentric 


366  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

circle,  which  in  strictness  ought  to  be  an  ellipse  not  differing  much  from  the 
circle. 

As  the  satellites  move  in  their  orbits  in  the  direction  of  the  arrows,  the 
curve  which  they  form  revolves  in  the  same  direction  with  a  velocity  1^  times 
that  of  the  ring. 

By  considering  these  figures,  and  still  more  by  watching  the  actual  motion 
of  the  ivory  balls  in  the  model,  we  may  form  a  distinct  notion  of  the  motions 
of  the  particles  of  a  discontinuous  ring,  although  the  motions  of  the  model  are 
circular  and  not  elliptic.  The  model,  represented  on  a  scale  of  one-third  in  figs. 
7  and  8,  was  made  in  brass  by  Messrs.  Smith  and  Ramage  of  Aberdeen. 

We  are  now  able  to  understand  the  mechanical  principle,  on  account  of 
which  a  massive  central  body  is  enabled  to  govern  a  numerous  assemblage  of 
satellites,  and  to  space  them  out  into  a  regular  ring;  while  a  smaller  central 
body  would  allow  disturbances  to  arise  among  the  individual  satelHtes,  and 
collisions  to  take  place. 

When  we  calculated  the  attractions  among  the  satellites  composing  the  ring, 
we  found  that  if  any  satellite  be  displaced  tangentially,  the  resultant  attraction 
will  draw  it  away  from  its  mean  position,  for  the  attraction  of  the  satellites  it 
approaches  will  increase,  while  that  of  those  it  recedes  from  will  diminish,  so  that 
its  equilibrium  when  in  the  mean  position  is  unstable  with  respect  to  tangential 
displacements ;  and  therefore,  since  every  satellite  of  the  ring  is  statically  unstable 
between  its  neighbours,  the  slightest  disturbance  would  tend  to  produce  coUisions 
among  the  satellites,  and  to  break  up  the  ring  into  groups  of  conglomerated 
sateUites- 

But  if  we  consider  the  dynamics  of  the  problem,  we  shall  find  that  this 
effect  need  not  necessarily  take  place,  and  that  this  very  force  which  tends 
towards  destruction  may  become  the  condition  of  the  preservation  of  the  ring. 
Suppose  the  whole  ring  to  be  revolving  round  a  central  body,  and  that  one 
satellite  gets  in  advance  of  its  mean  position.  It  will  then  be  attracted  forwards, 
its  path  will  become  less  concave  towards  the  attracting  body,  so  that  its  distance 
from  that  body  will  increase.  At  this  increased  distance  its  angular  velocity 
will  be  less,  so  that  instead  of  overtaking  those  in  front,  it  may  by  this  means 
be  made  to  fall  back  to  its  original  position.  Whether  it  does  so  or  not  must 
depend  on  the  actual  values  of  the  attractive  forces  and  on  the  angular  velocity 
of  the  ring.     When  the  angular  velocity  is  great  and  the  attractive  forces  small, 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  367 

the  compensating  process  will  go  on  vigorously,  and  the  ring  wiU  be  preserved. 
When  the  angular  velocity  is  small  and  the  attractive  forces  of  the  ring  great, 
the  dynamical  effect  wiU  not  compensate  for  the  disturbing  action  of  the  forces 
and  the  ring  ^vill  be  destroyed. 

If  the  satellite,  instead  of  being  displaced  forwards,  had  been  originally 
behind  its  mean  position  in  the  ring,  the  forces  would  have  pulled  it  backwards, 
its  path  would  have  become  more  concave  towards  the  centre,  its  distance  from 
the  centre  would  diminish,  its  angular  velocity  would  increase,  and  it  would 
gain  upon  the  rest  of  the  ring  till  it  got  in  front  of  its  mean  position.  This 
effect  is  of  course  dependent  on  the  very  same  conditions  as  in  the  former  case, 
and  the  actual  effect  on  a  disturbed  satellite  would  be  to  make  it  describe  an 
orbit  about  its  mean  position  in  the  ring,  so  that  if  in  advance  of  its  mean 
position,  it  first  recedes  from  the  centre,  then  falls  behind  its  mean  position  in 
the  ring,  then  approaches  the  centre  within  the  mean  distance,  then  advances 
beyond  its  mean  position,  and,  lastly,  recedes  from  the  centre  till  it  reaches  its 
starting-point,  after  which  the  process  is  repeated  indefinitely,  the  orbit  being 
always  described  in  the  direction  opposite  to  that  of  the  revolution  of  the 
ring. 

We  now  understand  what  would  happen  to  a  disturbed  satellite,  if  all  the 
others  were  preserved  from  disturbance.  But,  since  all  the  satellites  are  equally 
free,  the  motion  of  one  will  produce  changes  in  the  forces  acting  on  the  rest, 
and  this  will  set  them  in  motion,  and  this  motion  will  be  propagated  from  one 
satellite  to  another  round  the  ring.  Now  propagated  disturbances  constitute 
waves,  and  all  waves,  however  complicated,  may  be  reduced  to  combinations  of 
simple  and  regular  waves;  and  therefore  all  the  disturbances  of  the  ring  may 
be  considered  as  the  resultant  of  many  series  of  waves,  of  different  lengths,  and 
travelling  with  different  velocities.  The  investigation  of  the  relation  between 
the  length  and  velocity  of  these  waves  forms  the  essential  part  of  the  problem, 
after  which  we  have  only  to  split  up  the  original  disturbance  into  its  simple 
elements,  to  calculate  the  effect  of  each  of  these  separately,  and  then  to  combine 
the  results.  The  solution  thus  obtained  will  be  perfectly  general,  and  quite 
independent  of  the  particular  form  of  the  ring,  whether  regular  or  irregular  at 
starting.     §  14. 

We  next  investigated  the  effect  upon  the  ring  of  an  external  disturbing 
force.     Having  split  up  the  disturbing  force   into   components   of  the   same   type 


368  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

with  the  waves  of  the  ring  (an  operation  which  is  always  possible),  we  found 
that  each  term  of  the  disturbing  force  generates  a  "  forced  wave "  travelling  with 
its  own  angular  velocity.  The  magnitude  of  the  forced  wave  depends  not  only 
on  that  of  the  disturbing  force,  but  on  the  angular  velocity  with  which  the  dis- 
turbance travels  round  the  ring,  being  greater  in  proportion  as  this  velocity 
more  nearly  coincides  with  that  of  one  of  the  "free  waves"  of  the  ring,  "We 
also  found  that  the  displacement  of  the  satellites  was  sometimes  in  the  direction 
of  the  disturbing  force,  and  sometimes  in  the  opposite  direction,  according  to 
the  relative  position  of  the  forced  wave  among  the  four  natural  ones,  producing 
in  the  one  case  positive,  and  in  the  other  negative  forced  waves.  In  treating 
the  problem  generally,  we  must  determine  the  forced  waves  belonging  to  every 
term  of  the  disturbing  force,  and  combine  these  with  such  a  system  of  free 
waves  as  shall  reproduce  the  initial  state  of  the  ring.  The  subsequent  motion 
of  the  rmg  is  that  which  would  result  from  the  free  waves  and  forced  waves 
together.  The  most  important  class  of  forced  waves  are  those  which  are  pro- 
duced by  waves  in  neighbouring  rings.     §   15. 

We  concluded  the  theory  of  a  ring  of  satellites  by  tracing  the  process  by 
which  the  ring  would  be  destroyed  if  the  conditions  of  stability  were  not 
fulfilled.  We  found  two  cases  of  instability,  depending  on  the  nature  of  the 
tangential  force  due  to  tangential  displacement.  If  this  force  be  in  the  direction 
opposite  to  the  displacement,  that  is,  if  the  parts  of  the  ring  are  statically 
stable,  the  ring  will  be  destroyed,  the  irregularities  becoming  larger  and  larger 
mthout  being  propagated  round  the  ring.  When  the  tangential  force  is  in  the 
direction  of  the  tangential  displacement,  if  it  is  below  a  certain  value,  the 
disturbances  will  be  propagated  round  the  ring  without  becoming  larger,  and 
we  have  the  case  of  stability  treated  of  at  large.  If  the  force  exceed  this  value, 
the  disturbances  will  still  travel  round  the  ring,  but  they  will  increase  in  ampli- 
tude continually  till  the  ring  falls  into  confusion.     §  18. 

We  then  proceeded  to  extend  our  method  to  the  case  of  rings  of  different 
constitutions.  The  first  case  was  that  of  a  ring  of  satellites  of  unequal  size. 
If  the  central  body  be  of  suflScient  mass,  such  a  ring  will  be  spaced  out,  so  that 
the  larger  satellites  will  be  at  wider  intervals  than  the  smaller  ones,  and  the 
waves  of  disturbance  will  be  propagated  as  before,  except  that  there  may  be 
reflected  waves  when  a  wave  reaches  a  part  of  the  ring  where  there  is  a  change 
in  the  average  size  of  the  satellites.     §  19. 


ON    THE    STABILITY    OF     THE    MOTION    OF    SATURN  S     RINGS.  369 

The  next  case  was  that  of  an  annular  cloud  of  meteoric  stones,  revolving 
uniformly  about  the  planet.  The  avercige  density  of  the  space  through  which 
these  small  bodies  are  scattered  will  vary  with  every  irregularity  of  the  motion, 
and  this  variation  of  density  will  produce  variations  in  the  forces  acting  upon 
the  other  parts  of  the  cloud,  and  so  disturbances  will  be  propagated  in  this 
ring,  as  in  a  ring  of  a  finite  number  of  satellites.  The  condition  that  such  a 
ring  should  be  free  from  destructive  oscillations  is,  that  the  density  of  the 
planet  should  be  more  than  three  hundred  times  that  of  the  ring.  This  would 
make  the  ring  much  rarer  than  common  air,  as  regards  its  average  density, 
though  the  density  of  the  particles  of  which  it  is  composed  may  be  great. 
Comparing  this  result  with  Laplace's  minimum  density  of  a  ring  revolving  as 
a  whole,  we  find  that  such  a  ring  cannot  revolve  as  a  whole,  but  that  the  inner 
parts  must  have  a  greater  angular  velocity  than  the  outer  parts.     §  20. 

We  next  took  up  the  case  of  a  flattened  ring,  composed  of  incompressible 
fluid,  and  moving  with  uniform  angular  velocity.  The  internal  forces  here  arise 
partly  from  attraction  and  partly  from  fluid  pressure.  We  began  by  taking  the 
case  of  an  intinite  stratum  of  fluid  affected  by  regular  waves,  and  found  the  accurate 
values  of  the  forces  in  this  case.  For  long  waves  the  resultant  force  Is  in  the 
same  direction  as  the  displacement,  reaching  a  maximum  for  waves  whose 
length  is  about  ten  times  the  thickness  of  the  stratum.  For  waves  about  five 
times  as  long  as  the  stratum  is  thick  there  is  no  resultant  force,  and  for  shorter 
waves  the  force  is  in  the  opposite  direction  to  the  displacement.     §  23. 

Applying  these  results  to  the  case  of  the  ring,  we  find  that  it  will  be 
destroyed  by  the  long  waves  unless  the  fluid  is  less  than  -^  of  the  density  of 
the  planet,  and  that  in  all  cases  the  short  waves  will  break  up  the  ring  into 
small  satellites. 

Passing  to  the  case  of  narroiv  rings,  we  should  find  a  somewhat  larger 
maximum  density,  but  we  should  still  find  that  very  short  waves  produce  forces 
in  the  direction  opposite  to  the  displacement,  and  that  therefore,  as  already 
explained  (page  333),  these  short  undulations  would  increase  in  magnitude  without 
being  propagated  along  the  ring,  till  they  had  broken  up  the  fluid  filament  into 
drops.  These  drops  may  or  may  not  fulfil  the  condition  formerly  given  for  the 
stability  of  a  ring  of  equal  satellites.  If  they  fulfil  it,  they  will  move  as  a 
permanent  ring.  If  they  do  not,  short  waves  will  arise  and  be  propagated  among 
the  satellites,   with    ever  increasing  magnitude,  till  a   sufficient    number   of  drops 


370  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS. 

have    been    brought    into    collision,  so    as    to  unite   and  form  a  smaller  number  of 
larger  drops,   which  may  be  capable  of  revolving  as  a  permanent  ring. 

We  have  already  investigated  the  disturbances  produced  by  an  external 
force  independent  of  the  ring ;  but  the  special  case  of  the  mutual  perturbations 
of  two  concentric  rings  is  considerably  more  complex,  because  the  existence  of  a 
double  system  of  waves  changes  the  character  of  both,  and  the  waves  produced 
react  on  those  that  produced  them. 

We  determined  the  attraction  of  a  ring  upon  a  particle  of  a  concentric 
ring,  first,  when  both  rings  are  in  their  undisturbed  state ;  secondly,  when  the 
particle  is  disturbed  ;  and,  thirdly,  when  the  attracting  ring  is  disturbed  by  a 
series  of  waves.     §  26. 

We  then  formed  the  equations  of  motion  of  one  of  the  rings,  taking  in  the 
disturbing  forces  arising  from  the  existence  of  a  wave  in  the  other  ring,  and 
found  the  small  variation  of  the  velocity  of  a  wave  in  the  first  ring  as  dependent 
on  the  magnitude  of  the  wave  in  the  second  ring,  which  travels  with  it.     §  27. 

The  forced  wave  in  the  second  ring  must  have  the  same  absolute  angular 
velocity  as  the  free  wave  of  the  first  which  produces  it,  but  this  velocity  of 
the  free  wave  is  slightly  altered  by  the  reaction  of  the  forced  wave  upon  it. 
We  find  that  if  a  free  wave  of  the  first  ring  has  an  absolute  angular  velocity 
not  very  different  from  that  of  a  free  wave  of  the  second  ring,  then  if  both 
fi:ee  waves  be  of  even  orders  (that  is,  of  the  second  or  fourth  varieties  of  waves), 
or  both  of  odd  orders  (that  is,  of  the  first  or  third),  then  the  swifter  of  the 
two  free  waves  has  its  velocity  increased  by  the  forced  wave  which  it  produces, 
and  the  slower  free  wave  is  rendered  still  slower  by  its  forced  wave ;  and  even 
when  the  two  free  waves  have  the  same  angular  velocity,  their  mutual  action 
will  make  them  both  split  into  two,  one  wave  in  each  ring  travelling  faster, 
and  the  other  wave  in  each  ring  travelling  slower,  than  the  rate  with  which 
they  would  move  if  they  had  not  acted  on  each  other. 

But  if  one  of  the  free  waves  be  of  an  even  order  and  the  other  of  an  odd 
order,  the  swifter  free  wave  will  travel  slower,  and  the  slower  free  wave  will 
travel  swifter,  on  account  of  the  reaction  of  their  respective  forced  waves.  If 
the  two  free  waves  have  naturally  a  certain  small  difference  of  velocities,  they 
will  be  made  to  travel  together,  but  if  the  difference  is  less  than  this,  they 
will   again    split    into    two    pairs    of    waves,    one    pair    continually   increasing   in 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S     RINGS.  371 

magnitude  without  limit,  and  the  other  continually  diminishing,  30  that  one 
of  the  waves  in  each  ring  will  increase  in  violence  till  it  has  thrown  the  ring 
into  a  state  of  confusion. 

There  are  four  cases  in  which  this  may  happen.  The  first  wave  of  the 
outer  ring  may  conspire  with  the  second  or  the  fourth  of  the  inner  ring,  the 
second  of  the  outer  with  the  third  of  the  inner,  or  the  third  of  the  outer  with 
the  fourth  of  the  inner.  That  two  rings  may  revolve  permanently,  their  distances 
must  be  arranged  so  that  none  of  these  conspiracies  may  arise  between  odd 
and  even  waves,  whatever  be  the  value  of  m.  The  number  of  conditions  to 
be  fulfilled  is  therefore  very  great,  especially  when  the  rings  are  near  together 
and  have  nearly  the  same  angular  velocity,  because  then  there  are  a  greater 
number  of  dangerous  values  of  m  to  be  provided  for. 

In  the  case  of  a  large  number  of  concentric  rings,  the  stability  of  each  pair 
must  be  investigated  separately,  and  if  in  the  case  of  any  two,  whether  con- 
secutive rings  or  not,  there  are  a  pair  of  conspiring  waves,  those  two  rings  will 
be  agitated  more  and  more,  till  waves  of  that  kind  are  rendered  impossible  by 
the  breaking  up  of  those  rings  into  some  different  arrangement.  The  presence 
of  the  other  rings  cannot  prevent  the  mutual  destruction  of  any  pair  which 
bear  such  relations  to  each  other. 

It  appears,  therefore,  that  in  a  system  of  many  concentric  rings  there  will 
be  continually  new  cases  of  mutual  interference  between  different  pairs  of  rings. 
The  forces  which  excite  these  disturbances  being  very  small,  they  will  be  slow 
of  growth,  and  it  is  possible  that  by  the  irregularities  of  each  of  the  rings  the 
waves  may  be  so  broken  and  confused  (see  §  19),  as  to  be  incapable  of  mounting 
up  to  the  height  at  which  they  would  begin  to  destroy  the  arrangement  of  the 
ring.  In  this  way  it  may  be  conceived  to  be  possible  that  the  gradual  dis- 
arrangement of  the  system  may  be  retarded  or  indefinitely  postponed. 

But  supposing  that  these  waves  mount  up  so  as  to  produce  collisions  among 
the  particles,  then  we  may  deduce  the  result  upon  the  system  from  general 
dynamical  principles.  There  will  be  a  tendency  among  the  exterior  rings  to 
remove  further  from  the  planet,  and  among  the  interior  rings  to  approach  the 
planet,  and  this  either  by  the  extreme  interior  and  exterior  rings  diverging 
from  each  other,  or  by  intermediate  parts  of  the  system  moving  away  from  the 
mean   ring.     If  the   interior   rings  are  observed   to   approach    the  planet,   while   it 

47—2 


372  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

is  known  that  none  of  the  other  rings  have  expanded,  then  the  cause  of  the 
chancre  cannot  be  the  mutual  action  of  the  parts  of  the  system,  but  the  resistance 
of  some  medium  in  which  the  rings  revolve.  §  Si- 
There  is  another  cause  which  would  gradually  act  upon  a  broad  fluid  ring 
of  which  the  parts  revolve  each  with  the  angular  velocity  due  to  its  distance 
from  the  planet,  namely,  the  internal  friction  produced  by  the  slipping  of  the 
concentric  rings  with  different  angular  velocities.  It  appears,  however  (§  33), 
that  the  effect  of  fluid  friction  would  be  insensible  if  the  motion  were  regular. 

Let  us  now  gather  together  the  conclusions  we  have  been  able  to  draw 
from  the  mathematical  theory  of  various  kinds  of  conceivable  rings. 

We  found  that  the  stability  of  the  motion  of  a  solid  ring  depended  on 
so  delicate  an  adjustment,  and  at  the  same  time  so  unsymmetrieal  a  distribution 
of  mass,  that  even  if  the  exact  condition  were  fulfilled,  it  could  scarcely  last 
long,  and  if  it  did,  the  immense  preponderance  of  one  side  of  the  ring  would 
be  easily  observed,  contrary  to  experience.  These  considerations,  with  others 
derived  from  the  mechanical  structure  of  so  vast  a  body,  compel  us  to  abandon 
any  theory  of  solid  rings. 

We  next  examined  the  motion  of  a  ring  of  equal  satellites,  and  found  that 
if  the  mass  of  the  planet  is  sufficient,  any  disturbances  produced  in  the  arrange- 
ment of  the  ring  will  be  propagated  round  it  in  the  form  of  waves,  and  will  not 
introduce  dangerous  confusion.  If  the  satellites  are  unequal,  the  propagation  of 
the  waves  will  no  longer  be  regular,  but  disturbances  of  the  ring  will  in  this, 
as  in  the  former  case,  produce  only  waves,  and  not  growing  confusion.  Sup- 
posing the  ring  to  consist,  not  of  a  single  row  of  large  satellites,  but  of  a  cloud 
of  evenly  distributed  unconnected  particles,  we  found  that  such  a  cloud  must 
have  a  very  small  density  in  order  to  be  permanent,  and  that  this  is  inconsistent 
with  its  outer  and  inner  parts  moving  with  the  same  angular  velocity.  Supposing 
the  ring  to  be  fluid  and  continuous,  we  found  that  it  will  be  necessarily  broken 
up  into  small  portions. 

We  conclude,  therefore,  that  the  rings  must  consist  of  disconnected  particles ; 
these  may  be  either  solid  or  liquid,  but  they  must  be  independent.  The  entire 
system  of  rings  must  therefore  consist  either  of  a  series  of  many  concentric  rings, 
each  moving  with  its  own  velocity,  and  having  its  own  systems  of  waves,  or  else 
of  a  confused  multitude  of  revolving  particles,  not  arranged  in  rings,  and 
continually  coming  into  collision  with  each  other. 


ON    THE    STABILITY    OF    THE    MOTION     OF    rfATUKN  S     RINGS.  373 

Taking  the  first  case,  we  tbund  that  in  an  indefinite  number  of  possible 
cases  the  mutual  perturbations  of  two  rings,  stable  in  themselves,  might  mount 
up  in  time  to  a  destructive  magnitude,  and  that  such  cases  must  continually 
occur  in  an  extensive  system  like  that  of  Saturn,  the  only  retarding  cause  being 
the  possible  irregularity  of  the  rings. 

The  result  of  long-continued  disturbance  was  found  to  be  the  spreading 
out  of  the  rings  in  breadth,  the  outer  rings  pressing  outwards,  while  the  inner 
rings  press  inwards. 

The  final  result,  therefore,  of  the  mechanical  theory  is,  that  the  only  system 
of  rings  which  can  exist  is  one  composed  of  an  indefinite  number  of  unconnected 
particles,  revolving  round  the  planet  with  different  velocities  according  to  their 
respective  distances.  These  particles  may  be  arranged  in  series  of  narrow  rings, 
or  they  may  move  through  each  other  irregularly.  In  the  first  case  the  destruc- 
tion of  the  system  will  be  very  slow,  in  the  second  case  it  will  be  more  rapid, 
but  there  may  be  a  tendency  towards  an  arrangement  in  narrow  rings,  which 
may  retard  the  process. 

We  are  not  able  to  ascertain  by  observation  the  constitution  of  the  two 
outer  divisions  of  the  system  of  rings,  but  the  inner  ring  is  certainly  transparent, 
for  the  limb  of  Saturn  has  been  observed  through  it.  It  is  also  certain,  that 
though  the  space  occupied  by  the  ring  is  transparent,  it  is  not  through  the 
material  parts  of  it  that  Saturn  was  seen,  for  his  limb  was  observed  without 
distortion ;  which  shows  that  there  was  no  refraction,  and  therefore  that  the 
rays  did  not  pass  through  a  medium  at  all,  but  between  the  solid  or  liquid 
particles  of  which  the  ring  is  composed.  Here  then  we  have  an  optical  argument 
in  favour  of  the  theory  of  independent  particles  as  the  material  of  the  rings. 
The  two  outer  rings  may  be  of  the  same  nature,  but  not  so  exceedingly  rare 
that  a  ray  of  light  can  pass  through  their  whole  thickness  without  encounterino^ 
one  of  the  particles. 

Finally,  the  two  outer  rings  have  been  observed  for  200  years,  and  it  appears, 
from  the  careful  analysis  of  all  the  observations  by  M.  Struve,  that  the  second 
ring  is  broader  than  when  first  observed,  and  that  its  inner  edge  is  nearer  the 
planet  than  formerly.  The  inner  ring  also  is  suspected  to  be  approaching  the 
planet  ever  since  its  discovery  in  1850.  These  appearances  seem  to  indicate 
the  same  slow  progress  of  the  rings  towards  separation  which  we  found  to  be 
the  result  of  theory,    and  the    remark,   that  the    inner    edge    of  the  inner  ring  is 


374  ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN  S    RINGS. 

most  distinct,  seems  to  indicate  that  the  approach  towards  the  planet  is  less 
rapid  near  the  edge,  as  we  had  reason  to  conjecture.  As  to  the  apparent 
unchangeableness  of  the  exterior  diameter  of  the  outer  ring,  we  must  remember 
that  the  outer  rings  are  certainly  far  more  dense  than  the  inner  one,  and  that 
a  small  change  in  the  outer  rings  must  balance  a  great  change  in  the  inner 
one.  It  is  possible,  however,  that  some  of  the  observed  changes  may  be  due 
to  the  existence  of  a  resisting  medium.  If  the  changes  already  suspected  should 
be  confirmed  by  repeated  observations  with  the  same  instruments,  it  will  be 
worth  while  to  investigate  more  carefully  whether  Saturn's  Rings  are  permanent 
or  transitionary  elements  of  the  Solar  System,  and  whether  in  that  part  of 
the  heavens  we  see  celestial  immutability,  or  terrestrial  corruption  and  generation, 
and  the  old  order  giving  place  to  new  before  our  own  eyes. 


APPENDIX. 

On   the    Stability   of  the   Steady    Motion    of  a   Rigid   Body    about   a   Fixed    Centre   of  Force. 
By  Peofessor  W.  Thomson  {communicated  in  a  letter). 

The  body  will  be  supposed  to  be  symmetrical  on  the  two  sides  of  a  certain  plane 
containing  the  centre  of  force,  and  no  motion  except  that  of  parts  of  the  body  parallel 
to  the  plane  will  be  considered.  Taking  it  as  the  plane  of  construction,  let  G  (fig.  14) 
be  the  centre  of  gravity  of  the  body,  and  0  a  point  at  which  the  resultant  attraction  of 
the  body  is  in  the  line  OG  towards  G.  Then  if  the  body  be  placed  with  0  coinciding 
with   the   centre   of  force,   and   set  in   a   state   of  rotation   about   that   point   as   an  axis,  with 

an   angular   velocity   equal   to   A/Ajr.    (where    /    denotes   the   attraction    of    the   body   on   a 

unit  of  matter  at  0,  S  the  amount  of  matter  in  the  central  body,  M  the  mass  of  the 
revolving  body,  and  a  the  distance  OG),  it  will  continue,  provided  it  be  perfectly  undis- 
turbed, to  revolve  uniformly  at  this  rate,  and  the  attraction  Sf  on  the  moving  body  will 
be  constantly  balanced  by  the  centrifugal  force  oi'aM  of  its  motion. 

Let  us  now  suppose  the  motion  to  be  slightly  disturbed,  and  let  it  be  required  to 
investigate  the  consequences.  Let  X,  S,  Y,  be  rectangular  axes  of  reference  revolving 
uniformly  with  the  angular  velocity  (o,  round  S,  the  fixed  attracting  point.  Let  x,  y,  be 
the   co-ordinates   of  G  with   reference   to   these   axes,  and  let  XS,  YS  denote  the  components 


ON    THE    STABILITY    OF    THE    MOTION    OF    SATURN's    RINGS.  375 

of  the    whole    force   of  attraction    of  S   on    the   rigid    body.     Then    since   this   force   is   in  the 
line  through  S,  its  moment  round    G  is 

SYx-SXy; 
the   components    of    the    forces   on    the    moving   body    being   reckoned    as    positive   when   they 
tend   to   diminish   x   and   y   respectively.     Hence    if  k   denote    the   radius   of    gyration    of  the 
body   round    G,  and   if  <f>   denote   the   angle  which   OG  makes  with  SX  {i.e.  the  angle  GOK), 
the  equations  of  motion  are, 

In  the  first  place  we  see  that  one  integral  of  these  equations  is 

This  is  the  "equation  of  angidar  momentum." 

In  considering  whether  the  motion  round  S  with  velocity  co  when  0  coincides  with 
-S'  is  stable  or  unstable,  we  must  find  whether  every  possible  motion  with  the  same 
"  angular  momentum  "  round  S  is  such  that  it  will  never  bring  0  to  more  than  an  infinitely 
small  distance  from  S :  that  is  to  say,  we  must  find  whether,  for  every  possible  solution 
in  which  H  =  M  {ct"  +  k"")  o),  and  for  which  the  co-ordinates  of  0  are  infinitely  small  at  one 
time,  these  co-ordinates  remain  infinitely  small.  Let  these  values  at  time  t  be  denoted 
thus:  8^  =  ^,  and  NO='rj;  let  OG  be  at  first  infinitely  nearly  parallel  to  OX,  i.e.  let  <f> 
be  infinitely  small  (the  full  solution  will  tell  us  whether  or  not  <f)  remains  infinitely  small) ; 
then,  as  long  as  <f)  is  infinitely  small,  we  have 

x  =  a+  ^,     y  =  v  +  ^<^> 
and  the  equations  of  motion  have  the  forms 

31 


and  we  may  write  the  equation  of  angular  momentum  instead  of  the  third  equation. 

If  now  we  suppose  f  and  rj  to  be  infinitely  small,  the  last  of  these  equations  becomes 
{a'  +  k^)f^+2a>a^+af^=0 (a). 


376  ON    THE    STABILITY    OF    THE    MOTION    OF    SATUKN  S    RINGS. 

If  p  and   q   denote   the   components   parallel   and   perpendicular  to  OG  of  the   attraction 
of  the  body  on  a  unit  of  matter  at   S,   we  have 

X  =  pco?,^-q?,m4>  =  p,  and   F=psin^ +  5^003  ^=j3</>  4-^, 
since  q  and  ^  are  each  infinitely  small ;  and  if  we  put   V=  potential  at  S,  and 

then  p  =/-  a|  -  777,  q  =  -0v-  7^. 

If  we  make  these  substitutions  for  X  and    Y,  and  take  into  account  that 

.f=co'a^ (*). 

the  first  and  second  equations  of  motion  become 

g_2.^_„.f_2„af4(.f+„)=0 (0), 

A,2„|_„.,  +  „^4(^,+,«  =  0 W. 

Combining  equations  (a),  (c),  and  (tf),  by  the   same  method  as    that  adopted  in  the  text, 
we  find  that  the  differential  equation  in  ^,  7),  or  </>,  is  of  the  form 


d*u      ^d^u     ^ 


where  A  =  A;', 


C  =  a>*  (A;*  -  3a*)  +  «"  -^  {{a*  +  ^*)  (a  +  yS)  -  4a»y8}  +  {a'  +  Fj^^,  (a'yS  -  7). 

In  comparing  this  result  with  that  obtained  in  the  Essay,  we  must  put 

r^  for  a, 

R  for  M, 

B+S  for  S, 

L  for  o, 

Nt:  for  y8, 

Mr^  for  7. 


Tv^    7 


Fi^   Z 


VOL.  L  PLATE  V. 


Tig.  ^. 


Fi^    6. 


VOL.  L  PLATE  V, 


[From  the  Philosophical  Magazine  for  January  and  July,   I860.] 


XX.     Illustrations   of  the   Dynamical    Theory   of  Gases*. 

PART   L 

On  the  Motions  and  Collisions  of  Perfectly  Elastic  Spheres. 

So  many  of  the  properties  of  matter,  especially  when  in  the  gaseous  form, 
can  be  deduced  from  the  hypothesis  that  their  minute  parts  are  in  rapid  motion, 
the  velocity  increasing  with  the  temperature,  that  the  precise  nature  of  this 
motion  becomes  a  subject  of  rational  curiosity.  Daniel  Bemouilli,  Herapath, 
Joule,  Kronig,  Clausius,  &c.  have  shewn  that  the  relations  between  pressure, 
temperature,  and  density  in  a  perfect  gas  can  be  explained  by  supposing  the 
particles  to  move  with  uniform  velocity  in  straight  lines,  striking  against  the 
sides  of  the  containing  vessel  and  thus  producing  pressure.  It  is  not  necessary 
to  suppose  each  particle  to  travel  to  any  great  distance  in  the  same  straight 
line ;  for  the  effect  in  producing  pressure  \vill  be  the  same  if  the  particles 
strike  against  each  other ;  so  that  the  straight  line  described  may  be  very  short . 
M.  Clausius  has  determined  the  mean  length  of  path  in  terms  of  the  average 
distance  of  the  particles,  and  the  distance  between  the  centres  of  two  particles 
when  collision  takes  place.  We  have  at  present  no  means  of  ascertaining  either 
of  these  distances ;  but  certain  phenomena,  such  as  the  internal  friction  of  gases, 
the  conduction  of  heat  through  a  gas,  and  the  diffusion  of  one  gas  through 
another,  seem  to  indicate  the  possibility  of  determining  accurately  the  mean 
length  of  path  which  a  particle  describes  between  two  successive  collisions.  In 
order  to  lay  the  foundation  of  such  investigations  on  strict  mechanical  principles, 
I  shall  demonstrate  the  laws  of  motion  of  an  indefinite  number  of  small,  hard, 
and  perfectly  elastic  spheres  acting  on  one  another  only  during  impact. 

*  Read  at  the  Meeting  of  the  British  Association  at  Aberdeen,  Sei)tember  21,  1859. 
VOL.  I.  48 


378  ILLUSTRATIONS    OF    THE    DYNAMICAL    THEORY    OF    GASES. 

If  the  properties  of  such  a  system  of  bodies  are  found  to  correspond  to 
those  of  gases,  an  important  physical  analogy  will  be  established,  which  may 
lead  to  more  accurate  knowledge  of  the  properties  of  matter.  If  experiments 
on  gases  are  inconsistent  with  the  hypothesis  of  these  propositions,  then  our 
theory,  though  consistent  w^th  itself,  is  proved  to  be  incapable  of  explaining 
the  phenomena  of  gases.  In  either  case  it  is  necessary  to  follow  out  the 
consequences   of  the   hypothesis. 

Instead  of  saying  that  the  particles  are  hard,  spherical,  and  elastic,  we  may 
if  we  please  say  that  the  particles  are  centres  of  force,  of  which  the  action  is 
insensible  except  at  a  certain  small  distance,  when  it  suddenly  appears  as  a 
repulsive  force  of  very  great  intensity.  It  is  evident  that  either  assumption 
will  lead  to  the  same  results.  For  the  sake  of  avoiding  the  repetition  of  a 
long  phrase  about  these  repulsive  forces,  I  shall  proceed  upon  the  assumption 
of  perfectly  elastic  spherical  bodies.  If  we  suppose  those  aggregate  molecules 
which  move  together  to  have  a  bounding  surface  which  is  not  spherical,  then 
the  rotatory  motion  of  the  system  will  store  up  a  certain  proportion  of  the 
whole  vis  viva,  as  has  been  shewn  by  Clausius,  and  in  this  way  we  may 
accoimt  for  the  value  of  the  specific  heat  being  greater  than  on  the  more 
simple   hypothesis. 

On   the   Motion  and    Collision   of  Perfectly   Elastic   Spheres. 

Prop.  I.  Two  spheres  moving  in  opposite  directions  with  velocities*  inversely 
us  their  masses  strike  one  another;  to  determine  their  motions  after  impact. 

Let  P  and   Q  be   the   position   of  the   centres  at 
impact;    AP,   BQ  the   directions    and    magnitudes    of  ^-V  at 

the   velocities   before   impact;   Pa,   Qh   the   same   after  ^^^^^^^^ — j^ 

impact;  then,  resolving  the  velocities  parallel  and  per-  ^ 
pendicular  to  PQ  the  line  of  centres,  we  find  that 
tlie  velocities  parallel  to  the  line  of  centres  are  exactly 
reversed,  while  those  perpendicular  to  that  line  are 
luichanged.  Compounding  these  velocities  again,  we  find  that  the  velocity  of 
each  ball  is  the  same  before  and  after  impact,  and  that  the  directions  before 
and  after  impact  lie  in  the  same  plane  with  the  line  of  centres,  and  make  equal 
angles  with  it. 


ILLUSTRATIONS    OF    THE    DYNAMICAL    THEORY    OF    GASES.  379 

Prop.  11.  To  find  the  probability  of  the  direction  of  the  velocity  after 
impact   lying   between   given   limits. 

In  order  that  a  collision  may  take  place,  the  line  of  motion  of  one  of  the 
balls  must  pass  the  centre  of  the  other  at  a  distance  less  than  the  sum  of 
their  radii ;  that  is,  it  must  pass  through  a  circle  whose  centre  is  that  of  the 
other  ball,  and  radius  (s)  the  sum  of  the  radii  of  the  balls.  Within  this  circle 
every  position  is  equally  probable,  and  therefore  the  probability  of  the  distance 
from  the  centre  being  between  r  and  r  +  dr  is 

2rdr 
~7~' 

Now   let   <f>  be   the   angle   A  Pa  between  the  original  direction  and  the  directioii 
after  impact,  then  APN=^<f>,  and  7- =  5  sin  ^<^,  and  the  probabihty  becomes 

^  sin  6d^. 
The  area  of  a  spherical  zone  between  the  angles  of  polar  distance  <j>  and  <f)  +  d<f)  is 

27r  sin  (f)d<f> ; 

therefore   if  a>   be   any  small   area   on   the   surface  of  a  sphere,  radius  unity,  the 
probability  of  the  direction  of  rebound  passing  through  this  area  is 

to 

4:ir  * 

so   that   the   probability  is  independent   of  ^,   that   is,  all   directions   of  rebound 
are  equally  likely. 

Prop.  III.  Given  the  direction  and  magnitude  of  the  velocities  of  two 
spheres  before  impact,  and  the  line  of  centres  at  impact ;  to  find  the  velocities 
after  impact. 

Let  OA,  OB  represent  the  velocities  before  impact,  so  that  if  there  had  been 
no  action  between  the  bodies  they  would 
have  been  at  A  and  B  at  the  end  of  a 
second.  Join  AB,  and  let  G  be  their  centre 
of  gravity,  the  position  of  which  is  not 
affected  by  their  mutual  action.  Draw  GN 
parallel  to  the  line  of  centres  at  impact  (not 
necessarily   in   the    plane   AOB).     Draw   aGh 

in  the  plane  AGN,  making  NGa  =  NGA,  and  Ga=GA  and  Gb  =  GB;   then   by 

48—2 


380  ILLUSTRATIONS    OF    THE    DYNAMICAL    THEORY    OF    GASES. 

Prop.  I.  Ga  and  Gh  will  be  the  velocities  relative  to  G ;  and  compounding 
these  with  OG,  we  have  Oa  and  Oh  for  the  true  velocities  after  impact. 

By  Prop.  11.  all  directions  of  the  Une  aGh  are  equally  probable.  It  appears 
therefore  that  the  velocity  after  impact  is  compounded  of  the  velocity  of  the 
centre  of  gravity,  and  of  a  velocity  equal  to  the  velocity  of  the  sphere  relative 
to  the  centre  of  gravity,  which  may  with  equal  probability  be  in  any  direction 
whatever. 

If  a  great  many  equal  spherical  particles  were  in  motion  in  a  perfectly 
elastic  vessel,  collisions  would  take  place  among  the  particles,  and  their  velocities 
would  be  altered  at  every  collision;  so  that  after  a  certain  time  the  vis  viva 
will  be  divided  among  the  particles  according  to  some  regular  law,  the  average 
number  of  particles  whose  velocity  lies  between  certain  Umits  being  ascertainable, 
though  the  velocity  of  each  particle  changes  at  every  colUsion. 

Prop.  IV.  To  find  the  average  number  of  particles  whose  velocities  he 
between  given  limits,  after  a  great  number  of  collisions  among  a  great  number 
of  equal  particles. 

Let  N  be  the  whole  number  of  particles.  Let  x,  y,  z  be  the  components 
of  the  velocity  of  each  particle  in  three  rectangular  directions,  and  let  the  number 
of  particles  for  which  x  lies  between  x  and  x-hdx,  be  Nf{x)dx,  where  f{x)  is 
a  function  of  x  to  be  determined. 

The  number  of  particles  for  which  y  lies  between  y  and  y  +  dy  wUl  be 
Nf{y)dy;  and  the  number  for  which  z  Hes  between  z  and  z  +  dz  will  be  Nf(z)dz, 
where  /  always  stands  for  the  same  function. 

Now  the  existence  of  the  velocity  x  does  not  in  any  way  affect  that  of 
the  velocities  y  or  z,  since  these  are  all  at  right  angles  to  each  other  and 
independent,  so  that  the  number  of  particles  whose  velocity  lies  between  x  and 
x  +  dx,  and  also  between  y  and  y-{-dy,  and  also  between  z  and  z  +  dz,  is 

If  we  suppose  the  N  particles  to  start  from  the  origin  at  the  same  instant, 
then  this  wil)  be  the  number  in  the  element  of  volume  (dxdydz)  after  unit  of 
time,  and  the  number  referred  to  unit  of  volume  will  be 


ILLUSTRATIONS    OF    THE    DYNAMICAL    THEORY    OF    GASES.  381 

But   the   directions   of  the   coordinates   are  perfectly  arbitrary,  and  therefore  this 
number  must  depend  on  the  distance  from  the  origin  alone,  that  is 

f{x)f(y)f(z)  =  ^{^+y'  +  z% 

Solving  this  functional  equation,  we  find 

f{x)  =  Ce^'',         (^M  =  CV. 

If  we  make  A  positive,  the  number  of  particles  will  increase  with  the 
velocity,  and  we  should  find  the  whole  number  of  particles  infinite.  We  there- 
fore  make  A    negative   and   equal   to    — „ ,    so   that    the    number   between  x  and 

x  +  dx  is 

NCe'^'dx. 

Integrating  from  a:=— <»    toa;=-foo,we  find  the  whole  number  of  particles, 

aVTT 

1    -?: 

f[x)  is  therefore  /-e  "  . 

Whence  we  may  draw  the  following  conclusions  : — 

1st.  The  number  of  particles  whose  velocity,  resolved  in  a  certain  direction, 
lies  between  x  and  x  +  dx  is 

N^i'^'dx (1). 

2nd.     The  number  whose  actual  velocity  lies  between  v  and  v  +  dv  is 

]Sf-^^^e~^'dv (2). 

3rd.  To  find  the  mean  value  of  v,  add  the  velocities  of  all  the  particles 
together  and  divide  by  the  number  of  particles ;  the  result  is 

mean  velocity  =  -p- (3). 

Vtt 

4th.     To    find    the    mean    value    of    v;    add    all     the    values    together   and 

divide  by  N, 

mean  value  of  t;'  =  |a- (4). 

This  is  greater  than  the  square  of  the  mean  velocity,  as  it  ought  to  be. 


382  ILLUSTRATIONS    OF    THE    DYNAMICAL    THEORY    OF    GASES. 

It  appears  from  this  proposition  that  the  velocities  are  distributed  among 
the  particles  according  to  the  same  law  as  the  errors  are  distributed  among 
the  observations  in  the  theory  of  the  "  method  of  least  squares."  The  velocities 
i-ange  from  0  to  oo ,  but  the  number  of  those  having  great  velocities  is  com- 
paratively small.  In  addition  to  these  velocities,  which  are  in  all  directions 
equally,  there  may  be  a  general  motion  of  translation  of  the  entire  system  of 
particles  which  must  be  compounded  with  the  motion  of  the  particles  relatively 
to  one  another.  We  may  call  the  one  the  motion  of  translation,  and  the  other 
the  motion  of  agitation. 

Prop.  V.  Two  systems  of  particles  move  each  according  to  the  law  stated 
in  Prop.  IV. ;  to  find  the  number  of  pairs  of  particles,  one  of  each  system, 
whose  relative  velocity  lies  between  given  limits. 

Let  there  be  N  particles  of  the  first  system,  and  N'  of  the  second,  th